248 PD06 Abstracts

CP1

Efficient Optimization of Electrostatic InteractionsBetween Biomolecules

We will present a novel PDE-constrained optimizationstrategy for analyzing the optimality of electrostatic in-teractions between biomolecules. These interactions areimportant factors affecting binding affinity and specificity,and can be analyzed using linear continuum models. Theresulting optimization problems are well-posed but com-putationally expensive. Our method uses an implicit rep-resentation of the Hessian, in combination with precondi-tioned Krylov methods, to dramatically reduce the com-putational expense. We will discuss several applications inbiomolecule analysis and design.

Jacob WhiteResearch Lab. of ElectronicsMITwhite@mit.edu

Jaydeep BardhanElectrical Engineering & Computer ScienceMassachusetts Institute of Technologyjbardhan@mit.edu

Bruce TidorMITtidor@mit.edu

Michael AltmanDepartment of ChemistryMassachusetts Institute of Technologymaltman@mit.edu

CP1

Growth of a Spherical Tumor with a Necrotic Core:A Moving Boundary Simulation

After vascularization, a tumor may develop a necrotic core,characterized by an outer rim of proliferating cells and aninner core of dead cells. Research shows that between theseis a layer of quiescent cells which can begin dividing if en-vironmental conditions change. What follows is an exten-sion of previous work used to show the concentration ofnutrient within the interior of a tumor over time. At eachtime step, the core boundary and the tumor boundary arefree to increase or decrease as appropriate when comparedto certain threshold values, thereby simulating a movingboundary problem.

Carryn BellomoUniversity of Nevada, Las VegasDepartment of Mathematical Sciencecarryn.bellomo@unlv.edu

CP1

A Stochastic Model of the Tumor Growth for Dis-persed Cells Regime

A stochastic model is developed to describe the growth of aheterogeneous tumor for dispersed cells regime. The math-ematical model is a quasilinear stochastic partial differen-tial equation driven by a space-time white noise. The mainfeature of the model is that it takes into account randomindependent interactions between tumor cells, immune sys-tem cells and anticancer drugs. The existence of the weaksolutions and a comparison theorem are established. Some

applications are proposed.

Fejzi KolaneciUniversity of New York, Tiranafkolaneci@unyt.edu.al

CP1

Continuous Moran Process and the Diffusion ofGenes in Infinite Populations

We consider the so called Moran process with frequencydependent fitness given by a certain pay-off matrix. Forfinite populations, we show that the final state must behomegeneous, and show how to compute the fixation prob-abilities. Next, we consider the infinite population limit,and discuss the appropriate scalings for the drift-diffusionlimit. In this case, a degenerated parabolic PDE is formallyobtained that, in the special case of frequency independentfitness, recovers the celebrated Kimura equation in popula-tion genetics. We then show that the corresponding initialvalue problem is well posed and that the discrete modelconverges to the PDE model as the size of population goesto infinity. We also study some game-theoretic aspects ofthe dynamics and characterise the best strategies, in anappropriate sense.

Fabio ChalubDepartamento de MatematicaUniversidade Nova de Lisboachalub@cii.fc.ul.pt

Max O. SouzaDepartamento de Matematica AplicadaUniversidade Federal Fluminensemsouza@mat.uff.br

CP1

Computational Analysis Based on MathematicalModel for Biodegradation of Xenobiotic Polymers

Biodegradation processes of xenobiotic polymers are stud-ied numerically. The weight distribution with respect tothe molecular weight before and after biodegradation isintroduced into analysis based on a mathematical modeland an inverse problem is solved numerically to determinea biodegradation rate. Once the biodegradation rate isfound, an initial value problem can be solved numericallyto simulate the transition of the weight distribution, andthe applicability of the numerical result can be tested.

Masaji WatanabeGraduate School of Environmental ScienceOkayama Universitywatanabe@math.ems.okayama-u.ac.jp

Fusako KawaiResearch Institute for BioresourcesOkayama Universityfkawai@rib.okayama-u.ac.jp

CP1

Multiple-Spike Ground State Solutions of theGierer-Meinhardt Equations

In many biological pattern formations and some biochemi-cal reactions, an activator-inhibitor system of two reaction-diffusion equations serves as a mathematical model. The

PD06 Abstracts 249

typical Gierer-Meinhardt equations

At = dΔA−A+A2

H,

Ht = DΔH −H +A2,

where d/D << 1, A,H > 0 and A,H → 0 as | x |→ ∞, fea-ture two largely different diffusion coefficients and the es-sentially nonlocal nonlinearity. By the Lyapunov-Schmidtmethod, the maximum order estimate of the multiple spikenumbers of a ground state solution is found and the con-struction of such a solution is shown.

Yuncheng YouUniversity of South Floridayou@math.usf.edu

CP2

Particle Motion in the Circular, Restricted Three-Vortex Problem

Can a number of test particles traveling in different pe-riodic orbits in the circular, restricted three-vortex realmbe controlled such that they are placed in the same (equalperiod) orbit? Additionally, can the same or a differentcontroller be used to fix the relative positions of the testparticles such that a virtual or dynamically natural forma-tion can be established? It will be shown that the answerto both questions is, yes.

Ralph R. BasilioUniversity of Southern Californiarbasilio@usc.edu

Paul K. NewtonUniv Southern CaliforniaDept of Aerospace Engineeringnewton@spock.usc.edu

CP2

On Some Stochastic Fractional IntegrodifferentialEquations

The purpose of this paper is to study the integro-partialdifferential equation of fractional order:

∂αu(x, t)

∂tα−

∑|q|≤2m

aq(x)Dqu(x, t) = f1(u(x, t))+

∫ t

0

f2(u(x, s))dW (s),

with the nonlocal condition

u(x, 0) = ϕ(x) +

p∑k=1

cku(x, tk),

where 0 ≤ t1 < t2 < . . . < tp,x is an element of the n-dimensional Euclidean spaceRn, Dq = Dq1

1 . . . Dqnn ,

Dj =∂

∂xj, 0 < α ≤ 1,

q = (q1, . . . , qn) is an n dimensional multi-index, |q| =q1 + . . .+ qn and W (t) is standard Wiener process over thefiltered probability space Ω, F, Ft, P .It is supposed that

∑|q|=2m

aq(x)Dq is uniformly ellip-

tic on Rn . The existence of solutions of the consideredCauchy problem and some properties are studied under

suitable conditions on ϕ , the constants c1, . . . , cp, the func-tions aq, f1 and f2.

Mahmoud M. El-BoraiProf.of Math. Dep. of Mathematics Faculty of ScienceAlexandria University Alexandria Egyptm m elborai@yahoo.com

CP2

On the Stability of Some Stochastic DifferentialEquations

Stochastic Volterra equations of the form:

dx(t) = f(x(t))dt+

∫ t

0

K(t− s)x(s)ds dt+ g(x(t))dB(t),

are considered, where {B(t) : t ≥ 0} is standard one - di-mensional Brownian motion and the kernel K decreases tozero non-exponentially. We study the convergence rate tozero of the stochastic solutions of the considered equation.It is proved under suitable conditions that :

limt→∞

|x(t)|K(t)

= ∞, almost surely.

Khairia E. El-NadiProf of Mathematical Statistics Dep. of Math. Faculty ofScience Alexandria University Alexandria EgyptKhairia el said@Hotmail.com

CP2

Numerical Realizations of the Stochastic KdVEquation With and Without Damping

We investigate numerical simulations of an exact solutionof a stochastic Korteweg deVries equation under Gaussianwhite noise. We compare the expectation values of theexact solutions to theoretical expectation values and to thenumerical simulations of the stochastic Korteweg deVriesequation with and without damping. We find that typicallyon average the diffused soliton vanishes long before thetypically reported asymptotic limit.

Russell L. HermanUNC Wilmingtonhermanr@uncw.edu

CP2

First Passage Time Problem of a Time-DependentOrnstein-Uhlenbeck Process to a Moving Bound-ary

In this paper we derive the closed-form formula forthe first passage time density (FPTD) of a time-dependent Ornstein-Uhlenbeck process, i.e. the conven-tional Ornstein-Uhlenbeck process with time-dependentparameters, to a parametric class of moving boundariesby the method of images. We also apply the results todevelop a simple, efficient and systematic approximationscheme, namely the multistage approximation, to computetight upper and lower bounds of the FPTD through a fixedboundary.

Chi-Fai LoThe Chinese University of Hong Kongcflo@phy.cuhk.edu.hk

250 PD06 Abstracts

Cho-Hoi HuiHong Kong Monetary AuthorityResearch Departmentcho-hoi hui@hkma.org.hk

CP2

Convergence Analysis of Operator Upscaling forthe Acoustic Wave Equation

Wave propagation in a heterogeneous medium results inmodels involving multiple scales. Operator-based upscal-ing solves the problem on a coarse grid, but still retainssubgrid information and fine-scale input data. First, theproblem is solved for the subgrid component defined locallywithin each coarse block. Then, the subgrid solutions areused to augment the coarse-grid problem. We present con-vergence analysis for the method applied to the constantdensity, variable sound velocity acoustic wave equation.

Tetyana VdovinaDepartment of Mathematics and StatisticsUniversity of Maryland Baltimore Countytvdovin1@math.umbc.edu

Susan E. MinkoffUniversity of Maryland, Baltimore CountyDept of Mathematics and Statisticssminkoff@math.umbc.edu

CP3

Level Set Method Approaches to Dislocation Mod-els of Grain Boundaries

Grain boundaries in materials can be formally representedas arrays of dislocations lines. This perspective makes itpossible to study meso-scale processes, such as mechanismsfor grain boundary motion and interactions of dislocationswith grain boundaries. Using a level-set formalism fordislocation dynamics pioneered by our group, we inves-tigate several interesting grain boundary problems usingdislocation models. The computational efficiency requiredto study these models is provided by a parallel level setmethod software library that we have recently developed.

David SrolovitzPrinceton Universitysrol@princeton.edu

Yang XiangDepartment of MathematicsHKUSTmaxiang@ust.hk

Kevin T. ChuMechanical & Aerospace EngineeringPrinceton Universityktchu@princeton.edu

CP3

Free Boundary Interaction in Stefan Problem withUndercooling

Suppose that the Stefan problem with kinetic undercoolinghas a classical solution until the moment of contact of freeboundaries and the free boundaries have finite velocitiesuntil the moment of contact. Under these assumptions, weconstruct a smooth approximation of the global solutionof the Stefan problem with kinetic undercooling, which,

in the limit, gives the above classical solution until themoment of contact and becomes the solution of the heatequation after the contact. We study the properties of thisapproximation. In particular, we show that the velocities ofthe free boundaries become equal to each other in absolutevalue at the moment of contact and the temperature has ajump at the point of contact. The obtained analytical andnumerical results are compared.

Vladimir G. DanilovMoscow Technical University of Communications andInformaticProfessor, Head of Chairdanilov@miem.edu.ru

CP3

The Onset of Superconductivity at a Nor-mal/Superconducting Interface

We Study a modified model of Ginzburg and Landau thatconsiders superconducting electrons diffusing into a nor-mal material in contact with a superconductor. We assumethat each region occupy a half-space with a constant ap-plied field parallel to the interface. we show, if the normalconductivity of the superconductor is less than the conduc-tivity of the normal material then normal states are localminimizers for fields down to Hc2, which agrees with exper-imental observations that superconductivity is suppressedin this case. While when the conductivity of the supercon-ductor is larger than that for the normal material the onsetoccur at fields larger than Hc2 and less than Hc3.

Tiziana GiorgiNew Mexico State UniveristyDepartment of Mathematicstgiorgi@nmsu

Hala JadallahWestern Illinois UniversityDepartment of Mathematicsht-jadallah@wiu.edu

CP3

Homogenisation of Reaction, Diffusion and Interfa-cial Exchange in Porous Media with Evolving Mi-crostructure

Chemical degradation mechanisms of porous materials of-ten result in a change of the pore geometry. These effectscannot be captured by the standard periodic homogenisa-tion method due to the local evolution of the microscopicdomain. A mathematically rigorous approach is suggestedwhich allows the treatment of such problems. It makes useof a transformation to a stationary (periodic) reference do-main on which the homogenisation can be performed. Aphysical interpretation also allows the direct modelling ofthe transformed problem.

Malte A. Peter, Michael BohmCentre for Industrial MathematicsUniversity of Bremen, Germanympeter@math.uni-bremen.de,mbohm@math.uni-bremen.de

CP3

Solutions of the Smoluchowski Equation for Ne-

PD06 Abstracts 251

matic Polymers under Imposed Fields

We solve the Smoluchowski equation for rigid nematic poly-mers under imposed elongational flow, magnetic or electricfields. We show that: (1)The equation can be cast in ageneric form. (2) The steady state solution is of the Boltz-mann type, parametrized by material and two order pa-rameters. (3) The external field must be parallel to oneof the eigenvalues of the second moment tensor. A bifur-cation diagram of the order parameters is presented. Thesolution method is extended to dilute solutions under im-posed electric field. The first moment of the steady statesolution is shown parallel to the external field, under cer-tain conditions.

Qi WangFlorida State UniversityDepartment of Mathematicswang@math.fsu.edu

Sarthok K. SircarDepartment of MathematicsFlorida State Universityssircar@math.fsu.edu

Hong ZhouDepartment of Applied MathematicsNaval Postgraduate School, Monterey, CAhzhou@nps.edu

CP3

Dynamical Instability of Pinned Fundamental Vor-tices

We show dynamical instability of pinned fundamental ±1vortex solutions to the Ginzburg-Landau equations withexternal potentials in R2. For smooth and sufficiently smallexternal potentials, there exists a perturbed vortex solu-tion centered near each critical point of the potential. Per-turbed vortex solutions which are concentrated near min-ima (resp. maxima) are orbitally unstable (resp. stable).We consider both gradient and Hamiltonian flows.

Stephen GustafsonDepartment of MathematicsUniversity of British Columbiagustaf@math.ubc.ca

Fridolin TingLakehead UniversityDepartment of Mathematical Sciencesfting@lakeheadu.ca

CP4

The Taylor-Couette Problem in a DeformableCylinder

The Taylor-Couette problem is a fundamental example inbifurcation theory, and has been the subject of over 1500papers. This lecture treats a generalization where the rigidouter cylinder is replaced by a deformable (viscoelastic)cylinder. A steady solution of this liquid-solid interactionproblem can be found analytically; its linear stability isgoverned by a quadratic eigenvalue problem. The purposeof this lecture is to describe the analysis and computationof the spectrum.

David P. BourneUniversity of Maryland

dbourne@math.umd.edu

Stuart AntmanMathematics Department & IPSTUMCPssa@math.umd.edu

CP4

Floating Drops and Functions of Bounded Varia-tion

Three fluids are in equilibrium under the action of grav-itational and surface tension forces. The fluid with in-termediate density has a finite volume, and the motivat-ing example is that in which this volume forms a ’drop’floating on thhe most dense fluid wiith the least dense oneabove. In axisymmetric situations, for example when theother two fluids have infinite volume, this problem can beformulated as a free boundary problem for axisymmetriccapillary surfaces. The drop is bounded by a sessile dropand an inverted sessile drop and the exterior interface is anexterior capillary surface. This has been studied both the-oretically and numerically, but the general existence the-orem has proven difficult to obtain using these methods.We formulate here a variational problem using functions ofbounded variation and prove the existence of a solution.

Ray TreinenUniversity of Toledoray.treinen@gmail.com

Alan ElcratWichita State UniversityKansas, USAelcrat@math.twsu.edu

CP4

Explosive Instabilities In Rimming Flows

The linear stability of a thin film of viscous fluid on the in-side of a cylinder with horizontal axis, rotating about thisaxis is introduced in this paper. Both axial and azimuthalcomponents of the hydrostatic pressure gradient are takeninto account, which yield solutions that collapse in bothdimensions. Despite the existence of these ‘explosive’ in-stabilities, all solutions with harmonic dependence on theaxial variable and time (normal modes) are neutrally sta-ble. This type of instability has been described in previouspapers. However, no actual solution to describe the move-ment of the film of liquid through the cylinder has beenpresented. This paper will rectify this and examine theproperties of such a solution.

Sean M. LaceyUniversity of Limericksean.lacey@ul.ie

CP4

Ideal 2D Flow in a Domain with Holes and theSmall Obstacle Limit

We study the limiting behavior of incompressible, ideal flowin a 2D bounded domain with holes, when one of the holesbecomes small. The difficulty lies in describing and con-trolling the behavior of the potential part of the flow interms of vortex dynamics. This work is a natural exten-sion of [Iftimie et alli, CPDE 28 (2003) 349–370], whichstudied the same limit for flows in the exterior of a single

252 PD06 Abstracts

small obstacle.

Milton C. Lopes FilhoIMECC-UNICAMPmlopes@ime.unicamp.br

CP4

Analytic Solution of Rayleigh-Taylor Problem

We study the initial value problem for the classicalRayleigh-Taylor problems. If the initial data is analyticand satifies some other conditions, it is proved that uniqueanalytic solution exists locally in time. The analysisis based on a Nirenberg Theorem on abstract Cauchy-Kovalevsky problem in properly chosen Banach spaces.

Xuming XieMorgan State Universityxxie@jewel.morgan.edu

CP5

Equilibrium Ice Sheets Solve Variational Inequali-ties

The standard model for steady (shallow) ice sheet flow onan arbitrary bed with stress-dependent basal sliding andtemperature-dependent flow is shown to be a variationalinequality whose interior condition is the well-known andhighly nonlinear “ice sheet equation,” a nonlinear diffu-sion which generalizes the p-Laplacian equation. That is,such sheets solve a substantially-generalized p-Laplacianobstacle problem for p ≥ 2. These abstract problems arevariational inequalities for strictly monotone nonlinear op-erators on the Sobolev space W 1,p

0 (Ω). For these problemswe establish existence, an a priori bound, and continuityof the solution with respect to certain parameter fields.Uniqueness remains unproven in some cases. Finite ele-ment methods are considered.

Ed BuelerDept. of Mathematical StatisticsUniversity of Alaska Fairbanksffelb@uaf.edu

CP5

A Mathematical Model for Electromagnetic Wavesin Magnetized Media

The topic of this talk is the Landau-Lifschitz equationcoupled with Maxwell’s equations describing the electro-magnetic field in generally nonlinear magnetized medium.The Landau-Lifschitz equation is a first order ordinary dif-ferential equation for the magnetization field coupled toMaxwell’s equations for the electromagnetic field. Themain subjects are the existence, uniqueness and asymptoticbehavior of the solutions as well as certain small parameterlimits in this model.

Frank JochmannFakultaet II - Mathematik- Technische Universitaet Berlinjochmann@math.tu-berlin.de

CP5

Fractal Properties of Hele-Shaw Flow

Hele-Shaw flow in circular geometry leads to well-knowninstability of the frontline between two immiscible liq-

uids. Mathematically, flows of two liquids are describedby Laplace equations, with (curvature-dependent) bound-ary conditions. Though the numerical simulations allowone to solve the problem, there is no available techniquethat would describe the time-changing fractal properties ofthe resulting flows. Such novel approach for time-varyingdescription is proposed in the present work and well com-pared with experiment.

Igor Kliakhandler, Sofya ChepushtanovaDepartment of Mathematical SciencesMichigan Technological Universityigor@mtu.edu, svchepus@mtu.edu

CP5

An Approximated So-lution to the Two-Dimensional Lid-Driven CavityFlow, Using Adomian Decomposition Method andthe Vorticity-Stream Function Formulation

In this work, we present a reliable algorithm to solve thetwo-dimensional lid-driven cavity flow, using the AdomianDecomposition Method (ADM). The vorticity-stream func-tion formulation is used for the incompressible flow consid-ered here. The solution is calculated in the form of a serieswith easily computable coefficients. Also, numerical simu-lation, using finite difference method (FDM), is performedfor comparison purposes. This comparison shows consid-erably close agreements.

Mahmoud Najafikent state university ashtabuladrmnajafi@hotmail.com

M. Taeibi-RahniSharif University of Tech. Tehran Irantaeibi@sharif.ir

Khashayark AvaniSUT, Tehran Irankhaavani@gmail.com

CP5

Wave Propagation Through a Spatio-TemporalCheckerboard Structure

We consider propagation of waves through a spatio-temporal material structure with a checkerboard micro-geometry. The squares of the checkerboard are filled withalternating materials having equal impedance but differentphase speeds. Within certain parameter ranges, we observethe formation of distinct and stable limiting characteristicpaths that attract neighbouring characteristics after a fewtime periods. The average speed of propagation along thelimit cycles remains the same throughout certain ranges ofparameters of the microgeometry.

Suzanne L. WeekesWorcester Polytechnic Institute (WPI)sweekes@wpi.edu

CP5

Numerical Modeling of Fluid Mixing for Laser Ex-periments and Supernova

In collaboration with a team centered at U. Michigan andLLNL, we have conducted front tracking simulations foraxisymmetrically perturbed spherical explosions relevant

PD06 Abstracts 253

to supernovae as performed on NOVA laser experiments,with excellent agreement with experiments. We have ex-tended the algorithm and its physical basis for preshockinterface evolution due to radiation preheat. Our secondfocus is to study turbulent combustion in a type Ia super-nova (SN Ia) which is driven by Rayleigh-Taylor mixing.We have extended our front tracking to allow modeling ofa reactive front in SN Ia. Our 2d axisymmetric simulationsshow a successful level of burning.

Yongmin ZhangSUNY at Stony Brookyzhang@ams.sunysb.edu

James G. GlimmSUNY at Stony BrookDept of Applied Mathematicsglimm@ams.sunysb.edu

Paul DrakeUniversity of Michiganrpdrake@umich.edu

CP6

A PDE-Constrained Optimal Control Problem inImage Processing

We present a control problem motivated by mammographicimage processing. The underlying model involves a degen-erate parabolic PDE where the control enters in the diffu-sion coefficients and, implicitely, in the solution space. Wediscuss spaces of discontinuous functions adapted to thistype of control. We further show how a characterizationhelps in obtaining existence results for the simultaneoussolution of the control problem and the PDE.

Kristian Bredies, Dirk Lorenz, Peter MaassCenter for Industrial MathematicsUniversity of Bremenkbredies@math.uni-bremen.de, dlorenz@math.uni-bremen.de, pmaass@math.uni-bremen.de

CP6

Accelerated Inverse Analysis Using ParameterizedModel Order Reduction

One alternative to directly coupling solver and optimizerfor PDE constrained optimization problems is to use re-cently developed parameterized model reduction methodsto generate a parameterized low-order model and then usethe model in the optimizer. Such model-reduction basedoptimization methods have advantages in computationalefficiency, particularly when the extracted model is usedrepeatedly. We demonstrate that the method is very ef-fective for an inverse scattering problem associated withoptical inspection.

Luca DanielM.I.T.Research Lab in Electronicsluca@mit.edu

Jung Hoon Lee, Dimitry VasilyevMassachusetts Institute of Technologyjunghoon@mit.edu, vasilyev@mit.edu

Anne VithayathilNanobiosym, Inc.200 Boston Avenue, Suite 4700, Medford, MA 02055

anniev@alum.mit.edu

Jacob WhiteResearch Lab. of ElectronicsMITwhite@mit.edu

CP6

Existence of An Optimal Shape for a Pde Systemin a Free Air/Porous Domain: Application to Hy-drogen Fuel Cells

We consider a shape optimization problem related to a non-linear system of PDE describing the gas dynamics in a freeair - porous domain, including gas concentrations, temper-ature, velocity and pressure. The velocity and pressure aredescribed by Stokes and Darcy laws, while concentrationsand temperature are given by mass and heat conservationlaws. The system represents a simplified dry model of gasdynamics in channel and graphite diffusive layers in hydro-gen fuel cells. The model is coupled with the other partof domain through some mixed boundary conditions, in-volving nonlinearities, and pressure boundary conditions.Under some assumptions we prove that the system has a so-lution and that there exists a channel domain in the class ofLipschitz domains minimizing a certain functional measur-ing the membrane temperature distribution, total current,water vapor transport and channel inlet/outlet pressuredrop.

Arian NovruziUniversity of Ottawaarian.novruzi@mathstat.uottawa.ca

CP6

Hydrodynamic Limit to Hamilton-Jacobi Equa-tions

we study convolution-generated monotone cellular au-tomata(CA) and ordered transition rules. It turns out thatthose CA satisfy translation invariance and finite propaga-tion speed. We show that the hydrodynamic limit of theevolving occupied sets in such a CA is the level sets of theviscosity solution to a Hamilton-Jacobi equation, a firstorder geometric PDE

Minsu Songcombinatorial and computational mathematics centerPohang Univ. of Science and Technology, South Koreamsong@postech.ac.kr

CP6

Numerical Study of Optimal Control for NonlinearCahn-Hilliard Equations

Cahn-Hilliard (CH) equations are described a continuousmodel for phase transition in binary systems such as al-loy, glasses and polymer-mixtures. This work is to investi-gate numerical study for optimal control of CH equationsby the means of initial, boundary and distributed control.A semi-discrete algorithm is constructed for minimizationof quadratic optimal criteria using finite element method.The convergence and error estimate are deduced. Numer-ical demonstrations illustrated the efficiency of proposedparadigm.

Quan-Fang WangDepartment of Automation and Computer-Aided

254 PD06 Abstracts

EngineeringThe Chinese University of Hong Kongqfwang@acae.cuhk.edu.hk

CP7

Nonoscillatory Central Schemes on UnstructuredTriangular Grids for Hyperbolic Systems of Con-servation Laws

An extension to unstructured triangulations of Jiang andTadmor’s [SISC 19(6)] scheme will be proposed. Througha direction-insensitive reconstruction, the proposed schemeis of the correct order of accuracy in any direction, notjust along coordinate axes. Moreover, the use of an adap-tive limiter ensures the sharp resolution of discontinuitiesand convergence to the unique viscosity solution. Central-upwind extensions, in the spirit of Kurganov et al [SISC23(3)], will be considered.

Ivan ChristovDepartment of MathematicsTexas A&M Universitychristov@tamu.edu

Bojan PopovDepartment of MathematicsTexas A&M Universitypopov@math.tamu.edu

Peter PopovInstitute for Scientific ComputationTexas A&M Universtiyppopov@tamu.edu

CP7

A Hamiltonian Regularization of the BurgersEquation

We consider the following scalar equation:

ut + uux − α2utxx − α2uuxxx = 0, (1)

with α > 0. We may rewrite (1) as

vt + uvx = 0, (2)

wherev = u− α2uxx, (3)

One can think of the equation (2) as the inviscid Burgersequation,

vt + vvx = 0,

where the convective velocity in the nonlinear term is re-placed by a smoother velocity field u. This idea goes backto Leray (1934) who employed it in the context of theincompressible Navier-Stokes equation. Leray’s programconsisted in proving existence of solutions for his modifiedequations and then showing that these solutions converge,as α ↓ 0, to solutions of Navier-Stokes. We apply Leray’sideas in the context of Burgers equation. We show stronganalytical and numerical indication that (2)-(3) (or equiv-alently, (1)) represent a valid regularization of the Burgersequation. That is, we claim that solutions uα(x, t) of (1)converge strongly, as α→ 0, to unique entropy solutions ofthe inviscid Burgers equation. Interestingly, for all α > 0,the regularized equation possesses a Hamiltonian structure.We also study the stability of the traveling waves for equa-tion (1). These traveling waves consist of ”fronts”, whichare monotonic profiles that connect a left state to a right

state. The front stability results show that the regularizedequation (1) mirrors the physics of rarefaction and shockwaves in the Burgers equation.

Razvan FetecauStanford UniversityDepartment of Mathematicsvan@math.stanford.edu

CP7

Solutions to Coupled Systems of ConservationLaws

We discuss the construction of solutions to coupled sys-tems of nonlinear conservation laws. These equations arisefor example in models for traffic flow on road networks.The coupling is then due to the intersection of roads. Wederive necessary conditions including in particular the con-servation of all moments of the hyperbolic equation. Weconstruct solutions to the coupled system based on consid-erations on special Riemann problems posed at the roadintersections.

Michael HertyTechnische Universitaet KaiserslauternFachbereich Mathematikherty@mathematik.uni-kl.de

CP7

Transonic Regular Reflection for the Isentropic GasDynamics Equations

We consider a two-dimensional Riemann problem for theisentropic gas dynamics equations and derive regimes forwhich regular reflection can occur. We restrict our studyto transonic regular reflection and write the problem inself-similar coordinates. This change of variables leads toa mixed type system and a free boundary problem for aposition of the reflected shock and a subsonic state behindthe shock. We present preliminary ideas on analysis of thisproblem using the theory of second order elliptic equationsand the fixed point theory.

Suncica CanicDepartment of MathematicsUniversity of Houstoncanic@math.uh.edu

Barbara Lee KeyfitzFields Institute and University of Houstonblk@math.uh.edu

Katarina JegdicUniversity of Houstonkjegdic@math.uh.edu

CP7

High Resolution Total Variation DiminishingScheme for Linear System of Hyperbolic Conser-vation Laws

A conservative high-resolution TVD scheme for linear hy-perbolic systems is constructed using the flux limiter func-tion. In the present work the numerical flux function ofhigh-resolution scheme based on wave speed splitting isconstructed in such a way that it respects the physical hy-perbolicity property. Bounds are given for limiter func-tions to satisfy TVD property. Numerical results are given

PD06 Abstracts 255

for both 1-D and 2-D systems, which show high-resolutionnear points of discontinuity without introducing spuriousoscillations.

Ritesh KumarIndian Institute of Technology, Kanpurriteshkd@iitk.ac.in

Mohan K. KadalbajooIndian Institute of Technology KanpurKanpur-208016, Indiakadal@iitk.ac.in

CP7

A New Sticky Particle Method for Pressureless GasDynamics

I will first present a new sticky particle method for theone- and two-dimensional systems of pressureless gas dy-namics. The method is based on the idea of sticky par-ticles, which seems to work perfectly well for the modelswith point mass concentrations and strong singularity for-mations. In this method, the solution is sought in the formof a linear combination of delta-functions, whose positionsand coefficients represent locations, masses and momentaof the particles, respectively. The locations of the par-ticles are then evolved in time according to a system ofODEs, obtained from a weak formulation of the system ofPDEs. The particle velocities are approximated in a spe-cial way using a global conservative piecewise polynomialreconstruction technique over an auxiliary Cartesian mesh.This velocities correction procedure leads to a desired in-teraction between the particles and hence to clustering ofparticles at the singularities followed by the merger of theclustered particles into a new particle located at their cen-ter of mass.

In the two-dimensional case, the convergence of the pro-posed sticky particle method is still an open problem. I willshow that our particle approximation satisfies the originalsystem of pressureless gas dynamics in a weak sense, butonly within a certain error, which is rigorously estimated.I will also give an explanation why the relevant errors areexpected to diminish as the total number of particles in-creases.

Finally, I will demonstrate the performance of the newsticky particle method on a variety of one- and two-dimensional numerical examples and compare the obtainedresults with those computed by a high-resolution finite-volume scheme. The results obtained by the sticky particlemethod seem to be superior since the evolution of develop-ing discontinuities is accurately tracked and the developingdelta-shocks do not get smeared (this is the main advan-tage of using low-dissipative particle methods).

Alina ChertockNorth Carolina State UniversityDepartment of Mathematicschertock@math.ncsu.edu

Alexander KurganovTulane UniversityDepartment of Mathematicskurganov@math.tulane.edu

Yurii RykovKeldysh Institute of Applied MathematicsRussian Academy of Sciences

rykov@keldysh.ru

CP7

Entropy Solutions for a Triangular System of Con-servation Laws Modelling Three Phase Flow inPorous Media.

In this talk, we will consider a 2× 2 triangular (hence res-onant) system of conservation laws that models ”reduced”three phase flows in a porous medium where saturation ofone of the phases is independent of the other phases. A no-tion of entropy solutions is proposed and the entropy solu-tions are shown to form a L1 stable semi-group. Existenceof solutions is proved using a vanishing viscosity argumentand compensated compactness. Numerical schemes of theGodunov type are proposed and shown to converge to theentropy solution.

Kenneth Karlsen, NilsHenrik RisebroCentre of mathematics for applicationsDepartment of Mathematics, University of Oslokennethk@math.uio.no, nilshr@math.uio.no

Siddhartha MishraCenter of Mathematics for Applications,University of Oslo, Norway.siddharm@cma.uio.no

CP8

A Comparison of AB/BDI2 and AB/CN TimeStepping Schemes in a Chebychev-Tau SpectralNavier-Stokes Solver

The semi-implicit time stepping schemes: Adams-Bashforth/Backward-Differencing (AB/BDI2) and Adams-Bashforth/Crank-Nicolson (AB/CN) were implemented ina spectral methods based Navier-Stokes solver. The solverwas used for obtaining time dependent velocity and tem-perature fields for a natural convection in an inclined chan-nel problem. Contrary to the previous literature, for thisapplication, we have seen only a minor stability improve-ment when AB/BDI2 was used instead of AB/CN. Theimprovement was not enough to allow for a time step sizereduction but it was enough to decrease initial oscillationson the global solution. The comparative results and thecomparisons with the previous literature are presented interms of dimensionless groups.

Ilker Tari, Serkan Kasapoglu, Aykut ErdemMETUitari@metu.edu.tr, kserkan@metu.edu.tr,aykut@ceng.metu.edu.tr

CP8

Diagonalizing SimilarityTransformations for Variable-Coefficient Differen-tial Operators

Let L be a second-order self-adjoint variable-coefficient dif-ferential operator on Cp(2π). We show that using symboliccalculus and anti-differentiation operators, we can obtaina sequence of unitary similarity transformations that ap-proximately diagonalizes L, yielding analytical representa-tions of approximate eigenvalues and eigenfunctions. Eachtransformation has the form Lk+1 = U∗

kLkUk, where Uk

is the exponential of a skew-symmetric pseudodifferentialoperator. We will also discuss non-self-adjoint operators,higher-order operators, other boundary conditions, and

256 PD06 Abstracts

higher spatial dimension. Numerical results will be pre-sented.

Patrick GuidottiUniversity of California at Irvinegpatrick@math.uci.edu

James V. LambersStanford UniversityDepartment of Petroleum Engineeringlambers@stanford.edu

CP8

Four Corners: Analytic and Numerical Solution ofPoisson’s Equation with Discontinuous Coefficients

We present closed-form formulae and an efficient numericalmethod for Kellogg’s solution of Poisson’s equation withpiecewise constant coefficient on each quadrant of the unitsquare. The solution has a singular behaviour, rk, k < 1,r = distance from the junction point. We show how thismodel is a useful framework to analyze numerical methodsfor diffusion equations on Adaptive Mesh Refined (AMR),cell-centered, finite volume grids, common to CFD and oilreservoire simulations.

Oren E. LivneScientific Computing and Imaging InstituteUniversity of Utahlivne@sci.utah.edu

CP8

Intergrid Operators for Optimal Convergence ofMultigrid Algorithms

We study the effect of the interpolation and restriction op-erators on the convergence of multigrid algorithms for solv-ing linear PDEs. Using a modal analysis of a subclass ofthese systems, we determine how two groups of the modalcomponents of the error are filtered and mixed at each stepin the algorithm. We then show how to determine optimaltradeoffs between the convergence rate and computationalcomplexity for a given system.

Pablo Navarrete, Edward CoylePurdue Universitypnavarre@purdue.edu, coyle@purdue.edu

CP8

New Techniques for the Numerical Solution of El-liptic and Time Dependent Problems

The unified transform method of A. S. Fokas for solving lin-ear and certain monlinear PDEs, has led to important newdevelopments, regarding the analysis of various types ofPDE problems. This work is based on these developmentsand presents new techniques for the numerical solution ofelliptic boundary value problems, as well as time depen-dent problems. A comparative evaluation with existingwell established methods will also be provided.

Theodore S. Papatheodorou, Anastasia N. KandiliHPCLAB - Comp. Engin. & Inf. Dept.University of Patrasnkandylh@hpclab.ceid.upatras.gr,nkandylh@hpclab.ceid.upatras.gr

CP8

Monotonicity and Stability of Numerical Solutionsfor Obstacle Problems

Monotonicity and L∞-stability theorems are establishedfor a discrete obstacle problem which is defined by a piece-wise linear finite element discretization of a continuousproblem. These theorems extend the discrete maximumprinciple of Ciarlet from linear equations to obstacle prob-lems. As their applications, we establish a convergencetheorem for an iterative algorithm to solve discrete ob-stacle problems and extend Baiocchi’s and Nitsche’s L∞-estimates to a conforming full discrete obstacle problem.

Todd DupontUniversity of Chicagodupont@cs.uchicago.edu

Yongmin ZhangSUNY at Stony Brookyzhang@ams.sunysb.edu

CP9

Subspace Interpolation and Applications to Regu-larity for the Biharmonic Problem and the StokesSystems

We consider the biharmonic Dirichlet problem on a polygo-nal domain. New regularity estimates in terms of Sobolev-Besov norms of fractional order are proved. The analysisis based on new subspace interpolation results. We applyour results to establish regularity estimates for the Stokesand Navier-Stokes systems on polygonal domains.

Constantin BacutaUniversity of DelawareDepartment of Mathematicsbacuta@math.udel.edu

James H. BrambleTexas A&M UniversityCornel Universitybramble@math.tamu.edu

CP9

Some Nonlocal and Global Existence Results for aNonlinear Heat Equation with Critical Exponent

In this study, we consider the nonlinear heat equation{ut(x, t) = Δu(x, t) + up(x, t) in Ω × (0, T ),Bu(x, t) = 0 on ∂Ω × (0, T ),u(x, 0) = u0(x) in Ω,

with dirichlet and mixed boundary conditions, where Ω ⊂Rn is a smooth bounded domain and p = 1 + 2/n is thecritical exponent. For some initial condition u0 ∈ L1, weprove the nonlocal existence of the solution in L1 for themixed boundary condition. We also establish the globalexistence in L1+ε to the dirichlet problem, for any fixedε > 0 with ‖u0‖1+ε sufficiently small.

Canan CelikTOBB-ETUcanan.celik@etu.edu.tr

CP9

Positive Solutions to Nonlinear p-Laplace Equa-

PD06 Abstracts 257

tions with Hardy Potential in Exterior Domains

We study the existence and nonexistence of positive (su-per) solutions to the singular quasilinear p-Laplace equa-tion

−Δpu− μ

|x|p |u|p−2u =

C

|x|σ |u|q−1u

in exterior domains of RN (N ≥ 2). Here p ∈ (1,+∞)and μ ≤ CH , where CH is the critical Hardy constant. Weprovide a sharp characterization of the set of (q, σ) ∈ R2

such that the equation has no positive (super) solutions.

The proofs are based on the explicit construction of ap-propriate barriers and involve the analysis of asymptoticbehavior of super-harmonic functions associated to the p–Laplace operator with Hardy-type potentials, comparisonprinciples and an improved version of Hardy’s inequalityin exterior domains.

Sofya Lyakhova, Vitali Liskevich, Vitaly MorozSchool of MathematicsUniversity of Bristol, UKs.lyakhova@bris.ac.uk, V.Liskevich@bris.ac.uk,v.moroz@bris.ac.uk

CP9

Lewy-Hormander Nonexistence and Pseudospectra

In 1957 Lewy showed that linear PDE with smoothcoefficients can have no solutions, and soon thereafterHormander produced a general theory of such problemswhich was subsequenty generalized by Nirenberg, Treves,and others. In 2001 Zworski pointed out that the con-structions involved in this theory can be intrerpreted interms of pseudospectra. We will describe the connectionbetween Lewy-Hormander nonexistence and pseudospec-tra, and specifically, what are known as wave packet pseu-domodes of variable-coefficient differential operators.

Lloyd N. TrefethenOxford UniversityOxford University Compting LabLNT@comlab.ox.ac.uk

CP9

Entropy Solutions for the p(x)-Laplace Equation

We consider a Dirichlet problem in divergence form withvariable growth, modeled on the p(x)-Laplace equation.We obtain existence and uniqueness of an entropy solution.The proof is based on a priori estimates in Marcinkiewiczspaces with variable exponent. Joint work with ManelSanchon (CMUC, Portugal).

Jose Miguel UrbanoUniversidade de Coimbrajmurb@mat.uc.pt

CP9

Some Distributions Related to Boundary ValueProblems in Domains with Thin Non-Smooth Sin-gularities

The concept of wave factorization for an elliptic symbolwas introduced by the author. It is permitted to describethe structure of solutions (or solvability conditions) for amodel pseudodifferential equation in canonical non-smoothdomains of cone or wedge type. But canonical singularities

can be consisting of defferent dimensional components andalso non-smooth), and similar arguments lead to analogueresults in this case too.

Vladimir B. VasilyevDepartment of Mathematics and InformationTechnologiesBryansk State Universityvbv57@inbox.ru

CP10

Fast Explicit Operator Splitting Method forConvection-Diffusion Equations

Systems of convection-diffusion equations model a varietyof physical phenomena which often occur in real life. Com-puting the solutions of these systems, especially in theconvection dominated case, is an important and challeng-ing problem that requires development of fast, reliable andaccurate numerical methods. We propose a second-orderfast explicit operator splitting (FEOS) method based onthe Strang splitting. The main idea of the method is tosolve the parabolic problem via a discretization of the for-mula for the exact solution of the heat equation, whichis realized using a conservative and accurate quadratureformula. The hyperbolic problem is solved by a second-order finite-volume Godunov-type scheme. We provide atheoretical estimate for the convergence rate in the caseof one-dimensional systems of linear convection-diffusionequations with smooth initial data. Numerical convergencestudies are performed for one-dimensional nonlinear prob-lems as well as for linear convection-diffusion equationswith both smooth and nonsmooth initial data. We finallyapply the FEOS method to the one- and two-dimensionalsystems of convection-diffusion equations which model thepolymer flooding process in enhanced oil recovery. Our re-sults show that the FEOS method is capable to achieve aremarkable resolution and accuracy in a very efficient man-ner, that is, when only few splitting steps are performed.

Guergana PetrovaTexas A&M Universitygpetrova@math.tamu.edu

Alina ChertockNorth Carolina State UniversityDepartment of Mathematicschertock@math.ncsu.edu

Alexander KurganovTulane UniversityDepartment of Mathematicskurganov@math.tulane.edu

CP10

Global Existence and Uniqueness for a Model of theFlow of a Barotropic Fluid with Capillary Effects

We study the initial-value problem for a system of non-linear equations that models the flow of an inviscid,barotropic fluid with capillary stress effects. The systemincludes a hyperbolic equation for the velocity, and an al-gebraic equation (the equation of state). We prove theglobal-in-time existence of a unique, classical solution tothe system of equations. The key to the proof is a new L2

estimate for the density ρ.

Diane DennyTexas A&M University - Corpus Christi

258 PD06 Abstracts

ddenny@falcon.tamucc.edu

CP10

Nondegeneracy, from the Prospect of Wave-WaveRegular Interactions of a Gasdynamic Type

A parallel is constructed between Burnat’s ”algebraic” ap-proach [which uses a dual connection between the hodo-graph and physical characteristic details] and Martin’s ”dif-ferential” approach [centered on a Monge-Ampere typerepresentation] regarding their contribution to describingsome nondegenerate gasdynamic [one-dimensional, multi-dimensional] regular interaction solutions. • Some specificaspects of the multidimensional ”algebraic” description arebeforehand identified and classified with an admissibilitycriterion − selecting a ”genuinely nonlinear” type whereother (”hybrid”) types are formally possible.

Marina Ileana DinuPolytechnical University of Bucharest, Dept. of Math. IIIROMANIAmarinadinu@gmail.com

Liviu Florin DinuInstitute of Mathematics of the Romanian AcademyLiviu.Dinu@imar.ro, lfdinu2@gmail.com

CP10

Interconnection Between Quasilinear DifferentialEquations and Infinite Systems of Linear Pdes

We establish an amusing connection between infinite linearsystems of PDEs and quasilinear differential equations, in-cluding Hopf, Burger’s, Korteweg-de-Vriez equations, andother conservation laws. Such a connection enables toprove existence theorems and evaluate properties of solu-tions for infinite systems. Also, it provides the alterna-tive numerical method for solving quasilinear equations,based on numerical investigation of truncated linear sys-tems. The uniqueness of solutions can be obtained in spe-cial functional spaces generated by given infinite system.

Pavel B. DubovskiStevens Institute of Technologypdubovsk@stevens.edu

CP10

On the Reduced Description of Reactive EulerEquations of Detonation Theory

We describe a method for reducing reactive Euler equationsgoverning the dynamics of one-dimensional detonations toa nonlinear ordinary differential equation for the lead shockspeed. The reduced equation contains a single term thatneeds to be approximated which depends on spatial deriva-tives of pressure or particle velocity behind the lead shock.We discuss the properties of the reduced equation in con-nection with pulsating instability of one-dimensional deto-nation and also show how the equation can be incorporatedinto direct numerical simulation of the reactive Euler equa-tions.

Aslan R. KasimovDepartment of MathematicsMassachusetts Institute of Technologyaslan@mit.edu

CP11

The Stability of Couette Flow in a Toroidal Mag-netic Field*

The stability of the hydromagnetic Couette flow is investi-gated when a constant current is applied along the axis ofthe cylinders. It is shown that if the resulting toroidal mag-netic field depends only on this current, no linear instabilityto axisymmetric disturbances is possible. *This material isbased upon work supported in part by the U. S. Depart-ment of Energy under Grant No. DE-FG02-05ER25666 (toI.H.)

Isom Herron, Fritzner SolimanRensselaer Polytechnic Instituteherroi@rpi.edu, solimf@rpi.edu

CP11

Asymptotic Behavior Near Transition Fronts forEquations of Cahn-Hilliard Type

We consider the asymptotic behavior of perturbations ofstanding wave solutions arising in evolutionary PDE ofCahn–Hilliard type in one space dimension. Under the as-sumption of spectral stability, described in terms of an ap-propriate Evans function, we develop detailed asymptoticsfor perturbations from standing wave solutions, establish-ing phase-asymptotic orbital stability for initial perturba-tions decaying with appropriate algebraic rate.

Peter HowardDepartment of MathematicsTexas A&M Universityphoward@math.tamu.edu

CP11

Decay of Solutions to Some Nonlinear Wave Equa-tions with Dissipation

We study asymptotic behavior of solutions to the Cauchyproblem of some nonlinear wave equations of the form

ut + P (u)x +K(u) = 0, (∗)

where P and u are real-valued functions, K is de-

fined by Kv(ξ) = k(ξ)v(ξ) and the circumflexes de-note Fourier transforms. The Heat equation, the gen-eralized Burgers equations, the generalized Korteweg-deVries-Burgers equations, the generalized regularized longwave-Burgers equations, and the generalized Benjamin-Ono-Burgers equations are particular forms of equation(*). The asymptotical representations for large time arestudied in detail.

Jerry BonaUniversity of Illinois at Chicagobona@math.uic.edu

Thanasis FokasDepartment of Applied Mathematics and TheoreticalPhysicsUniversity of Cambridge, UKt.fokas@damtp.cam.ac.uk

Laihan LuoRichard Stockton College of New Jerseyrobert.luo@stockton.edu

PD06 Abstracts 259

CP11

Stability and Attractors for Quasi-Steady Model ofCellular Flames

We discuss the recently introduced model of quasi-steadydevelopment of cellular flames. The Quasi-Steady equa-tion (QSE) demonstrates the same basic instability mech-anism as its better-known counterparts, the Kuramoto-Sivashinsky equation (KSE), and the Burgers-Sivashinsky(BS) equation (the latter being just a linearly forced Burg-ers equation). Namely, a linear stability analysis revealsthe long-wave destabilization, which is suppressed, for thesmall wavelengths, by the dominant dissipative principalterm. In a sense, QSE is intermediate between BSE andKSE, as its dispersion relation coincides with that for BSEfor short waves, and is identical to that of KSE (up to thefourth order in k) for long waves. Note that, while BShas trivial dynamics, similarly to KSE, QSE demonstratevery rich dynamical behavior. We demonstrate that QSEpossesses a universal absorbing set, and a compact attrac-tor of finite Hausdorff dimension, and give an estimate onit. This is a joint work with Claude-Michel Brauner andJosephus Hulshof.

Michael FrankelDepartment of MathematicsIUPUImfrankel@math.iupui.edu

Victor RoytburdDepartment of Mathematical SciencesRPIroytbv@rpi.edu

CP11

Asymptotic Behavior of FitzHugh-Nagumo Equa-tions

We study the asymptotic behavior of the FitzHugh-Nagumo system on unbounded domains. It is shown thatthe system has a compact global attractor which containstraveling wave solutions. We also investigate the limitingbehavior of the attractors as a parameter ε → 0. It isproved that the limiting system has no global attractor,but all attractors for the perturbed system with positivebut small ε are contained in a common compact set.

Bixiang WangDepartment of Mathematics, New Mexico Mining andTechnologybwang@nmt.edu

MS1

Uniqueness of Weak Solutions of the Navier-StokesEquations of Multidimensional, Compressible Flow

We prove uniqueness and continuous dependence on ini-tial data of weak solutions of the Navier-Stokes equationsof compressible flow in two and three space dimensions.The solutions we consider may display codimension-onediscontinuities in density, pressure, and velocity gradient,and consequently are the generic singular solutions of thissystem. The key point of the analysis is that solutionswith minimal regularity are best compared in a Lagrangeanframework; that is, we compare the instantaneous statesof corresponding fluid particles in two different solutionsrather than the states of different fluid particles instan-taneously occupying the same point of space-time. Esti-mates for H−1 differences in densities and L2 differences

in velocities are obtained by duality from bounds for thecorresponding adjoint system.

David HoffIndiana Universityhoff@indiana.edu

MS1

An Equation for the Formation of Step Bunches inthe Morphological Evolution of Nano-scale CrystalSurface Structures Below the Roughening Transi-tion

Lagrangian coordinates continuum equations for surfaceevolution are obtained from a nano-scale description (stepinteraction equations). The surface consists of interactingsteps, moving by adatom diffusion on terraces; and attach-ment and detachment at steps. Diffusion-limited (DL) andattachment-detachment limited (ADL) kinetics, are con-sidered, for axially symmetric surfaces. The ADL equa-tions capture step bunching. The bunches appear fromthe interaction between a (destabilizing) negative diffusion(step-line tension effects) and a stabilizing nonlinear diffu-sion (step-step interactions). The local bunch-dynamics isdescribed by a simple nonlinear equation combining theseeffects.

Rodolfo R. RosalesMassachusetts Inst of TechDepartment of Mathematicsrrr@math.mit.edu

Dionisios MargetisDepartment of MathematicsMassachusetts Institute of Technologydio@math.mit.edu

MS1

On Phase Transition Dynamics

A multidimensional model is introduced for the dynamicsof a binary mixture of compressible fluids, which undergopossible phase transition due to chemical reaction. Themodel presented here can accommondate various physicalcontexts, namely liquid- liquid, gas-liquid, gas-gas phaseequilibria, the evolution of gaseous stars, the dynamics ofsemiconductors and others. The model is formulated bythe Navier Stokes equations in Euler coordinates, whichis now expressed by the conservation of mass, the balanceof momentum and entropy and the species conservationequation. This system takes now a new form due to thechoice of rather complex constitutive relations, which areable to accommondate the binary character of the mixture.The existence of globally defined weak solution is obtainedby the use of weak convergence methods and compactnessarguments in the spirit of Feireisl and P.L. Lions.

Konstantina TrivisaUniversity of MarylandDepartment of Mathematicstrivisa@math.umd.edu

MS1

On Compressible Navier-Stokes Equation

We consider the isentropic compressible Navier-Stokesequation with density dependent viscosity coefficients. Wefocus our study in the multivariable case when those coef-

260 PD06 Abstracts

ficients vanish on vacuum. In particular we prove the sta-bility of weak solutions of this problem using new kind ofentropy inequalities which involve gradients of the density.This leads to new regularity on the density which, surpris-ingly, holds only for degenerated viscosity coefficients.

Alexis VasseurUniversity of Texas at Austinvasseur@math.utexas.edu

Antoine MelletDepartment of MathematicsUniversity of Texas-Austinmellet@math.utexas.edu

MS2

Controllability of a Coupled System that ModelsCochlear Dynamics

The standard 2-dim cochlea model consists of a one-dimelastic structure (modeling the basilar membrane) sur-rounded by an incompressible 2-dim fluid within a 2-dimcochlear cavity. The dynamics are typically driven by apressure differential across the basilar membrane transmit-ted through the round and oval windows (a portion of theboundary of the cochlea). First we describe a highly ide-alized model in which the basilar membrane is modeled asan infinite array of oscillators and the fluid is describedby Laplace’s equation. In this idealized setting we showthat the coupled system is approximately controllable withcontrol acting on an arbitrary open set of the basilar mem-brane. If the basilar membrane has longitudinal membrane(string) elasticity, then exact controllability can be proved.

Scott HansenIowa State UniversityDepartment of Mathematicsshansen@iastate.edu

Isaac ChepkwonyIowa State Universityichep@iastate.edu

MS2

Uniform Stability of a Nonlinear Plate-BeamModel

The question of uniform stability is considered for a modelcomprised of a nonlinear vonKarman plate coupled witha nonlinear beam equation. Stability properties of linkedstructures composed of multiple elastic elements give rise toan abundance of mathematical challenges. When a struc-ture is composed of interconnected elastic elements of dif-ferent dimension, the behavior becomes much harder toboth predict and to control. (Joint work with GuenterLeugering.)

Mary Ann HornVanderbilt Universitymary.ann.horn@Vanderbilt.Edu

MS2

Tissue Modeling and its Influence on Shear Stiff-ness Imaging

We discuss tissue properties and their importance in build-ing an elastic model that will yield accurate algorithmsand shear stiffness images. Experimental design, elastic,

viscoelastic, and anisotropic models will be discussed. Im-ages with laboratory data will be presented.

Kui Lin, Jens KleinRensselaer Polytechnic Institutelink@rpi.edu, kleinj4@rpi.edu

Jeong-Rock YoonDepartment of Mathematical SciencesClemson Universityjryoon@clemson.edu

Daniel RenziDepartment of Mathematical SciencesRensselaer Polytechnic Instituterenzid@rpi.edu

Joyce R. McLaughlinRensselaer Polytechnic InstDept of Mathematical Sciencesmclauj@rpi.edu

MS3

Propagation of Fronts in Porous Media Combus-tion

Gaseous detonation is a phenomenon with a very compli-cated dynamics. The mathematical study of the qualitativebehavior of solutions of the full problem is out of reach atthe moment. Explosions in inert porous media provide amodel that is realistic, rich and suitable for a mathemat-ical analysis. We will present some recent mathematicalresults concerning the long time behavior of solutions ofthis model. In particular, initiation of detonation, forma-tion of the detonation front, its stability and propagationlimits will be discussed.

Peter GordonNJITpeterg@njit.edu

MS3

A Variational Approach to Front Propagation inInfinite Cylinders

Gradient reaction-diffusion-advection systems arise in thecontext of modeling the kinetics of phase transitions, popu-lation dynamics and combustion. These systems are knownto exhibit a variety of non-trivial spatio-temporal behav-iors, most notably the phenomenon of propagation andtraveling waves. We introduce a variational formulation forthe traveling wave solutions in cylindrical geometries withtransverse potential flow, which allows us to construct acertain class of traveling wave solutions and establish theirmonotonicity, asymptotic decay and uniqueness. These so-lutions are special in a sense that they are characterizedby a non-generic fast exponential decay ahead of the waveand play an important role in propagation phenomena forthe initial value problem. We also construct an area-typefunctional that gives a matching upper and lower boundfor the propagation speed in the sharp reaction zone limitfor weakly curved fronts.

Cyrill B. MuratovNew Jersey Institute of TechnologyDepartment of Mathematical Sciencesmuratov@njit.edu

Matteo Novaga

PD06 Abstracts 261

Department of MathematicsUniversity of Pisanovaga@dm.unipi.it

MS3

Diffusion and Mixing in Fluid Flow

We study enhancement of diffusive mixing on a compactRiemannian manifold M by a fast incompressible flow.We provide a sharp characterization of the class of flowsthat make the deviation of the solution of an equationφt + Au · ∇φ = Δφ from its average arbitrarily small inan arbitrarily short time, provided that the flow amplitudeA is large enough. The necessary and sufficient conditionon such flows is that the dynamical system associated withthe flow has no eigenfunctions in the Sobolev spaceH1(M).In particular, we find that weakly mixing flows always en-hance dissipation in this sense.

Leonid RyzhikUniversity of ChicagoDepartment of Mathematicsryzhik@math.uchicago.edu

P. ConstantinDepartment of MathematicsUniversity of Chicagoconst@math.uchicago.edu

A. Kiselev, A. ZlatosDepartment of MathematicsUniversity of Wisconsinkiselev@math.wisc.edu, andrej@math.wisc.edu

MS3

Spreading of Combustion in the Presence of StrongCellular Flows with Gaps

Recently Fannjiang, Kiselev and Ryzhik showed that cel-lular flows are capable of quenching large flames, providedthe amplitude of the flow is also large. We show that thisphenomenon does not occur when flow cells are separatedby narrow buffer zones where advection is absent.

Andrej ZlatosDepartment of MathematicsUniversity of Wisconsinzlatos@math.wisc.edu

MS4

The Impact of a Van Der Waals Type InterfacialEnergy on Dimensional Reduction

The asymptotic behavior of an elastic thin film penalizedby a van der Wals type interfacial energy is investigatedwhen both its thickness and the magnitude of the addi-tional energy vanish in the limit. Keeping track of bothmid-plane and out of plane deformations (through the in-troduction of the Cosserat vector), the resulting behaviorstrongly depends upon the ratio between thickness and in-terfacial energy. Non-locality is conjectured for a criticalvalue of that ratio.

Gilles FrancfortL.P.M.T.M., Universite Paris-Nord, Villetaneuse, Francefrancfor@galilee.univ-paris13.fr

MS4

Transport in Molecular Motor Systems

The molecular motor systems which govern much of thetransport in eukaryotes may be thought of as governed bya dissipation principle involving conformational changes in-duced by chemical reactions and potentials. The result inthe simplest cases is a weakly coupled system of evolutionequations. To the mind’s eye, the transport process is anal-ogous to a biased coin toss. We describe how this intuitionmay be confirmed by a careful analysis of the cooperativeeffect among the conformational changes and the poten-tials. This is joint work with Stuart Hastings and BryceMcLeod.

David KinderlehrerCarnegie Mellon Universitydavidk@andrew.cmu.edu

MS4

Surfactants in Foam Stability: A Phase FieldModel

The role of surfactants in stabilizing the formation of bub-bles in foams is studied using a phase-field model. Themodel involves a van der Walls-Cahn-Hilliard-type energywith an added term accounting for the interplay betweenthe presence of a surfactant density and the creation of in-terfaces. In agreement with experimentation, it is provedthat the surfactant segregates to the interfaces. This iswork in collaboration with Massimiliano Morini and Va-leriy Slastikov.

Irene FonsecaCarnegie Mellon UniversityCenter for Nonlinear Analysisfonseca@cmu.edu

Massimiliano MoriniSISSATrieste,Italymorini@sissa.it

MS4

Critical Points of Onsager Model of Nematic PhaseTransition

We study Onsagers model of isotropic-nematic phase tran-sitions with orientation parameter on a sphere. We con-sider symmetric Maier-Saupe potential and prove the ax-ial symmetry and derive explicit formulae for all criticalpoints, thus obtaining their complete classification.

Valeriy SlastikovUniversity of Warwick, Warwick, UKvaleriy@maths.warwick.ac.uk

MS4

Limit of the Non Self-dual Chern-Simons-Higgs En-ergy

We present some results on the gamma limit of the Chern-Simons-Higgs energy functional, far from self duality. Weuse a mixture of methods used in the study of relatedGinzburg-Landau energies.

Daniel SpirnUniversity of Minnesotaspirn@math.umn.edu

262 PD06 Abstracts

MS5

Wiener’s Criterion for Uniqueness of Solutions tothe Elliptic and Parabolic Problems in Domainswith Noncompact Boundaries

In this talk I will introduce a notion of regularity of infinityfor the elliptic and parabolic equations and will formulatea necessary and sufficient condition for the existence ofa unique bounded solution to the classical and parabolicDirichlet problems in arbitrary open sets with noncompactboundaries. The criterion is the exact analogue of Wienerstest for boundary regularity of harmonic and thermic func-tions and demonstrates the thinness of the exterior of adomain at infinity.

Ugur G. AbdullaDepartment of Mathematical SciencesFlorida Institute of Technologyabdulla@fit.edu

MS5

Harnack Estimates for Non-negative Solutions ofQuasi-linear Degenerate Parabolic Equations withMeasurable Coefficients

The classical Harnack estimates for non-negative solutionsof the heat equations do not hold for solutions of degener-ate equations (counterexample will be given). The estimateholds however in an ”intrinsic form”. The novelty of theinvestigation is that it continues to hold for quasi-lineardegenerate equations with merely measurable coefficients.This will be connected with the issue of local Holder con-tinuity of solutions of such degenerate equations. The ap-proach is entirely different from the classical approach ofMoser, in that it dispenses with BMO, coverings and cross-over, and it is only based on measure-theoretical ideas.

Vincenzo VespriDepartment of MathematicsUniversity of Florencevespri@math.unifi.it

Ugo P. GianazzaDepartment of MathematicsUniversity of Paviagianazza@imati.cnr.it

Emmanuele DiBenedettoDepartment of MathematicsVanderbilt University, Nashvilleem.diben@vanderbilt.edu

MS5

Continuity of Solutions of Nonlinear FiltrationEquations

We consider local bounded solutions of

β(u)t =

n∑i,j=1

Di(aij(x, t)Dj(u))

where the coefficients aij are uniformly elliptic, boundedand measurable functions and β is a C1-piecewise func-tion with at least a linear growth. If s0 is an irregularpoint for β, we assume β to be locally concave in a properright neighborhood of s0 and locally convex in a properleft neighborhood of the same point. By using suitable De-Giorgi’s techniques, we have the following alternative:

(1) either u is continuous and its modulus of continuitycan be given in a quantitative way,

(2) or in cylinders Q of length β(s)s

we have that

β′(s)sβ(s)

≤ C1|{(x, t) ∈ Q such that u(x, t) ≤ s}|

|Q| ≤ C2(log s)−1

This latter inequality implies the regularity of the solutionfor a large class of functions β and therefore we are able toextend DeGiorgi’s theory to suitable non power-like singu-larities.

Vincenzo VespriDepartment of MathematicsUniversity of Florencevespri@math.unifi.it

Ugo P. GianazzaDepartment of MathematicsUniversity of Paviagianazza@imati.cnr.it

MS5

On a Boundary Harnack Inequality for p-HarmonicFunctions

Let D ⊂ Rn be a bounded Lipschitz domain, x ∈ ∂D, andB(x, r) = {y ∈ Rn : |y − x| < r}. Given p, 1 < p < ∞, letu, v be positive weak solutions to the p Laplacian in D ∩B(x, r) with u = v = 0, continuously on ∂D ∩B(x, r). Weshow there exists c depending only on p, n and the Lipschitzconstant for D such that u/v ≤ c in B(x, r/2)∩D. We alsodiscuss Holder continuity of u/v.

Kaj NystromUmea University, Swedenkaj.nystrom@math.umu.se

John LewisUniversity of Kentuckyjohn@ms.uky.edu

MS5

The Neumann Problem for the Infinity-Laplacian

We consider the ∞-Laplacian with Neuman boundary con-ditions. We study the limit as p → ∞ of solutions of−Δpup = 0 in a domain Ω with |Dup|p−2∂up/∂ν = g on∂Ω. We obtain a natural minimization problem that is ver-ified by a limit point of {up} and a limit problem that issatisfied in the viscosity sense. Hoewever, contrary to theDirichlet case, there is no uniqueness for viscosity solutionsof the limit problem. It turns out that the limit variationalproblem is related to the Monge-Kantorovich mass transferproblem when the measures are supported on ∂Ω.

Juan J. ManfrediDepartment of MathematicsUniversity of Pittsburghmanfredi@pitt.edu

MS6

A Kinetic Transport and Reaction Model for Pro-cess Models in Microelectronics Manufacturing

The manufacturing of microelectronics devices involve gasflow at various total pressures, characterized by the Knud-sen number as relevant dimensionless group. We introduce

PD06 Abstracts 263

a kinetic transport and reaction model and simulator basedon a system of time-dependent linear Boltzmann equationsthat is valid over a wide range of Knudsen numbers. Wepresent results that demonstrate the ability of the model toanalyze the behavior of the system and to provide uniqueinsights into non-equilibrium effects.

Timothy CaleRensselaer Polytechnic InstituteFocus Center: Interconnections for Gigascale Integrationcalet@rpi.edu

Matthias K. GobbertUniversity of Maryland, Baltimore CountyDepartment of Mathematics and Statisticsgobbert@math.umbc.edu

MS6

Discontinuous Galerkin Approximation of theBoltzmann-Poisson System

We analyze the problem of numerically solving linear vari-ants of the space inhomogeneous multidimensional linearBoltzmann equation for charged transport. Determinis-tic methods can be very accurate for problems in whichthe solution may or may not be far from thermodynamicalequilibrium and high accuracy is required. Additionally,they yield specific details of transient solutions and con-sequently can be more efficient than the traditional prob-abilistic Monte Carlo techniques for the computation oftransients. We employ a discontinuous Galerkin finite ele-ment method to preserve local conservation properties andexploit the advantages of these methods for the compu-tations of Boltzmann-Poisson problems with large dopingprofiles, such as in the modeling of semiconductor devicesor collisional plasma. Theoretical and numerical results arediscussed.

Jennifer ProftThe University of Texas at Austinproft@math.utexas.edu

MS6

A WENO Algorithm for the Radiative Transferand Ionized Sphere at Reionization

In this presentation, we will show that the algorithm basedon the weighted essentially nonoscillatory (WENO) schemewith anti-diffusive flux corrections can be used as a solverof the radiative transfer equations. This algorithm is highlystable and robust for solving problems with both disconti-nuities and smooth solution structures. We tested this codewith the ionized sphere around point sources. It shows thatthe WENO scheme can reveal the discontinuity of the ra-diative or ionizing fronts as well as the evolution of photonfrequency spectrum with high accuracy on coarse meshesand for a very wide parameter space. This method wouldbe useful to study the details of the ionized patch given byindividual source in the epoch of reionization. We demon-strate this method by calculating the evolution of theion-ized sphere around point sources in physical and frequencyspaces. It shows that the profile of the fraction of neutralhydrogen and the ionized radius are sensitively dependenton the intensity of the source.

Long-Long FengPurple Mountain ObservatoryChinafengll@pmo.ac.cn

Li-Zhi FangDepartment of PhysicsUniversity of Arizonafanglz@physics.arizona.edu

Jingmei QiuBrown UniversityDivision of Applied Mathematicsjqiu@dam.brown.edu

Chi-Wang ShuDivision of Applied MathematicsBrown Universityshu@dam.brown.edu

MS6

Deterministic Solver for Non-linear BoltzmannDissipative Equations

We present a kinetic solver for Boltzmann energy dissipa-tive equations based on spectral methods. We compute ex-ample of states associated to homogeneous cooling for gran-ular flows and dynamically scaled solutions to elastic gasesmodels in the presence of a thermostat. These models havestationary or self-similar states that correspond to non-equilibrium statistical states (NESS). In particular we showthat our calculations resolve long time dynamics to prob-ability distribution functions with finite or infinity energy,unbounded at the origin and power tails, which are self-similar solutions to energy dissipative Maxwell Moleculesmodels.

Sri Harsha TharkabhushanaUniversity of Texas, Austinharsha@math.utexas.edu

Irene M. GambaDepartment of Mathematics and ICESUniversity of Texasgamba@math.utexas.edu

MS7

Extracting s-wave Scattering Lengths from BECDynamics

See MS1 in under annual meeting (AN06 abstracts.)

David S. HallDepartment of PhysicsAmherst Collegedshall@amherst.edu

MS7

Multi-Component Vortex Solutions in CoupledNLS Equations

See MS1 in under annual meeting (AN06 abstracts.)

Dmitry PelinovskyMcMaster University, Canadadmpeli@math.mcmaster.ca

MS7

Bose-Einstein Condensates in Optical Lattices and

264 PD06 Abstracts

Superlattices

See MS1 in under annual meeting (AN06 abstracts.)

Mason A. PorterCalifornia Institute of TechnologyDepartment of Physicsmason@caltech.edu

MS7

Derivation of the Gross-Pitaevskii Equation for Ro-tating Bose Gases

See MS1 in under annual meeting (AN06 abstracts.)

Robert SeiringerDepartment of MathematicsPrinceton Universityrseiring@Math.Princeton.edu

MS8

Variational Systems of Wave Equations

We consider systems of wave equations that are governedby variational principles whose Lagrangians are quadraticfunctions of the derivatives of the wave-field with coeffi-cients depending on the wave-field itself. We introducenotions of genuine nonlinearity and linear degeneracy forsuch equations that are analogous to, but different from,the corresponding notions for hyperbolic systems of conser-vation laws, and we derive a new weakly nonlinear asymp-totic equation for waves that lose genuine nonlinearity in asystem. We use this equation it to study the propagationof ‘twist’ waves in a massive director field.

Giuseppe AliConsiglio Nazionale delle Ricerche, Napoli, Italyali@na.iac.cnr.it

John HunterDepartment of MathematicsUC Davishunter@math.ucdavis.edu

MS8

The L1 Well-Posedness for Steady Supersonic EulerFlow Past a Curved Wedge by Wave Front TrackingMethod

We study the L1 well-posedness for two-dimensional steadysupersonic Euler flows past a Lipschitz wedge. The wedgeperturbs the flow and the waves reflect after interactingwith the strong shock front or the boundary. After study-ing the Lyapunov functional between two solutions anddealing with the boundary difficulty, we prove that thefunctional decreases in the flow direction. Therefore, theL1 stability is established, so is the uniqueness of the solu-tions obtained by the wave front tracking method.

Tianhong LiStanford Universitythli@math.Stanford.EDU

Gui-Qiang ChenNorthwestern UniversityUSAgqchen@math.northwestern.edu

MS8

A Mixed Type Problem for Two-dimensional WaveInteractions

The wave interactions in two-dimensional Riemann prob-lems of gas dynamics will be discussed. A degenerate Gour-sat problem will be studied. The solution in the hyperbolicregion up to the sonic curve is obtained.

Dehua WangUniversity of PittsburghDepartment of Mathematicsdwang@math.pitt.edu

MS8

Periodic Solutions of Euler’s Equations

We consider the interactions of shocks in the p-system. Weobtain bounds for incident shocks of arbitrary strength,arbitrarily close to the vacuum. We obtain bounds for thestates themselves, and for the wave strengths. We willdiscuss these in the context of global solutions with BVinitial data.

Robin YoungDepartment of MathematicsUniversity of Massachusetts Amherstyoung@math.umass.edu

MS9

Homogenization of Strongly Local Dirichlet Formsin Perforated Media

We investigate the homogenization for strongly localDirichlet forms in perforated media giving an extension ofthe results known for Laplace operator. The main difficul-ties arrive from the absence of a good notion of periodicityand of extension properties for ”regular” domains.

Marco BiroliDept. MathematicsPolytechnic Institute Milanomarbir@mate.polimi.it

Nicoletta TchouUniversity of Rennes 1nicoletta.tchou@univ-rennes1.fr

MS9

Undulations and Fluctuations

The issue of computing the flux of a fluctuating vectorfield across an undulating surface is commonly approachedby exploring the limits within which the formula estab-lished under blanket smoothness hypotheses is still valid,the general wisdom being that the wigglier the surface, thesmoother the field has to be (and vice versa). I argue herethat the dim no man’s land lying just beyond these limitsis well worth being penetrated, even though cautiously.

Antonio DiCarloDept. of Studies on Structures University “Roma Tre”adicarlo@mac.com

MS9

A Constructive Approach to Some Fractal Trans-

PD06 Abstracts 265

mission Problems

We describe a constructive approach to certain fractaltransmission problems for second order elliptic or parabolicoperators. The transmission condition on the layer is ofsecond order. The fractal layer is obtained in the limit ofpolyhedral (pre-fractal) surfaces, by self-similar iterationat small scales. From an analytic point of view, the mainproblem to be studied is the convergence of the solutionsobtained with the pre-fractal layers when the geometry be-comes fractal. The main difficulty in this study is the jumpof the geometric dimension of the layer, which is two for allpre-fractal approximations and intermediate between twoand three for the limit layer. In other words, while theapproximating layers have finite two-dimensional area, thelimit layer is non rectifiable with infinite area. We firstconsider the elliptic case. We will describe the variationalapproach consisting in proving the convergence of suitableenergy forms in the sense of homogeneization and we willalso describe some extensions to the heat equation, by rely-ing on the convergence of the related semigroups. In orderto obtain these convergence properties, we have to intro-duce suitable renormalization factors.

Maria Rosaria LanciaDept. Applied MathematicsUniversity of Roma La Sapienzalancia@dmmm.uniroma1.it

MS9

On Some Second Order Transmission Problems

We describe some transmission problems across fractal lay-ers imbedded in Euclidean domains. There is a wide lit-erature dealing with transmission problems of this kind,which arise naturally in various fields: e.g., in electrostat-ics, magnetostatics, hydraulic fracturing and in the studyof absorption or irrigation properties. In many of these ap-plications, one is interested in considering layers that pos-sess much greater conductivity or permeability than thesurrounding space. By referring, for example, to heat con-duction, the heat flow is then absorbed by the layer andstarts diffusing within it much more efficiently. This leadsto a jump of the normal derivatives across the two sidesof the layer and this jump acts as a source term for thediffusion within the layer. In our talk we shall describesome model examples of second order elliptic transmissionproblems with highly conductive layers. The transmissioncondition is also of second order, what is unusual in the caseof second order operators. Moreover, the condition has animplicit character, since the source term of the layer equa-tion - the jump of the normal derivatives - is not amongthe data of the problem, but depends on the solution itselfin the surrounding space. The layers we consider - say, ina domain of R3 - have an additional unusual characteris-tic: they are of fractal type, with a (Hausdorff) dimensionintermediate between 2 and 3. Layers of this type can beexpected to enhance the layer effects described above, byabsorbing more energy from the surrounding space. Wepresent some regularity and numerical results, in the aimof better understanding the analytical problems which arisewhen fractal and Euclidean structures mutually interact.

Maria Agostina VivaldiDept. Applied MathematicsUniversity of Roma La Sapienzavivaldi@dmmm.uniroma1.it

MS10

The Stability of Fronts in Cahn–Hilliard TypeEquations

The stability of stationary fronts in Cahn–Hilliard typeequations is important to the understanding of the evolu-tion of fronts in the sharp interface limit. In this talk wediscuss a method for establishing the asymptotic stabilitythat is based on a detailed analysis of dissipation at thefronts. This leads to a system of non-linear inequalitieswhich may be used to prove convergence. The method isfairly robust, and may be used to derive conditions for sta-bility in a range of setting. The work presented here isjoint work with M. Carvalho and E. Orlandi

Maria CarvalhoUniversity of Lisbon and Georgia Techmcarvalh@math.gatech.edu

Enza OrlandiUniversity of Rome IIIorlandi@mat.uniroma3.it

Eric A. CarlenGeorgia Techcarlen@math.gatech.edu

MS10

Thin-film Equations: Blow-up, the Cauchy Prob-lem, Solutions and Countable Spectra of Asymp-totic Patterns

We consider the fourth-order thin film equation in R ×(0, T ) with an unstable second-order term

ut = −(|u|nux)xxx − (|u|p−1u)xx, n ∈ [0,3

2), p > 1.

We show that at the critical exponent p = p0 = n+3, thereexists a countable spectrum of blow-up patterns, where thefirst one is expected to be stable. We study both non-negative solutions of the free-boundary zero contact an-gle problem and oscillatory sign changing solutions of theCauchy problem. Some other aspects concerning the cor-rect functional setting of the Cauchy problem are discussedwhere the limit n→ 0 plays a key role. We also investigatethe uniqueness question for the free-boundary problem.

Viktor GalaktionovDept. of Math. SciencesUniversity of Bathvag@maths.bath.ac.uk

MS10

Blow-up for Quasilinear Parabolic Equations withThin-Film-Diffusion Like Operators

We consider higher-order thin film equations with lower-order terms of reaction type. Using an extension of thenonlinear capacity method, we calculate the critical Fujitaexponents establishing precise parameter intervals whereno global nontrivial solutions are available.

Stanislav PohozhaevSteklov Institute of MathematicsRAS, Moscowpohozaev@mi.ras.ru

266 PD06 Abstracts

MS10

The Degenerate Cahn-Hilliard Equation and Wet-ting

We consider

ut + (un(uxxx + uβux))x = 0, x ∈ (0, L), t > 0,

ux = unuxxx = 0, x = 0, L, t > 0,

which generalizes the (one-sided) deep quench Cahn-Hilliard equation and constitutes also a thin film equa-tion in the unstable regime, when β < 2 and n is suitablyrestricted. Using energy estimates corresponding to theequation under consideration as well as thin film entropyarguments, we prove global existence of non-negative en-tropy solutions. Using a generalised Stampacchia lemma,estimates on the rate of propagation of the degeneracy sethave also been obtained

Andrey ShishkovInstitute of Applied Math. and Mechanics of NAS ofUkraine,Donetsk, Ukraineshishkov@iamm.ac.donetsk.us

Amy Novick-CohenDepartment of MathematicsTechnion, Haifa, Israelamync@tx.technion.ac.il

MS11

Telephone-cord Instabilities in Thin Smectic Cap-illaries

Telephone-cord patterns have been recently observed insmectic liquid crystal capillaries. In this talk we analysethe effects that may induce them. As long as the capil-lary keeps its linear shape, we show that a nonzero chiralcholesteric pitch favors the SmA*-SmC* transition. How-ever, neither the cholesteric pitch nor the presence of anintrinsic bending stress are able to give rise to a curvedcapillary shape. The key ingredient for the telephone-cordinstability is spontaneous polarization. The free energyminimizer of a spontaneously polarized SmA* is attainedon a planar capillary, characterized by a nonzero curva-ture. More interestingly, in the SmC* phase the combinedeffect of the molecular tilt and the spontaneous polarizationpushes towards a helicoidal capillary shape, with nonzerocurvature and torsion.

Paolo BiscariDipartimento di MatematicaPolitecnico di Milanopaolo.biscari@polimi.it

MS11

Phase Transition from Chiral Nematic TowardSmectic Liquid Crystals

The Chen-Lubensky energy is used to investigate phasetransitions from chiral nematic to smectic C* and smecticA* liquid crystal phases. We consider a liquid crystallinematerial confined between two parallel plates, where thedimensions of the material are assumed to be large rela-tive both to the width of a smectic layer and the mate-rial’s chiral pitch. We take boundary conditions so thatthe smectic phase melts at the plates’ surfaces and provethe existence of energy minimizers. Then under the phys-ically observed assumption that the Frank elasticity con-stants become large near a phase transition, we establish

estimates for the transition region separating phases. Inparticular we derive analytic estimates proving that chiral-ity lowers the transition temperature regime above whichminimizers are nematic and below which minimizers are ina smectic phase.

Sookyung JooInstitute for Mathematics and its Applications, UMNsjoo@ima.umn.edu

MS11

Steady Poiseuille Flows of Nematic Liquid Crystalsat Large

We consider a system of nonlinear second order ordinarydifferential equations modeling Poiseuille flow of liquidcrystals with variable degree of orientation, at the limitof large Ericksen number. The system is singularly per-turbed and degenerate, and as a result the solutions arehighly oscillatory. We obtain the relations satisfied by theYoung measures generated by sequences of weak solutions,and show that the persistent oscillations are encoded inthe Young measure generated by the molecular alignmentvariable φ. The effective equations consist of the isotropicNewtonian flow equation, supplemented by algebraic mo-mentum relations for the Young measure generated by φ.The latter relations impose restrictions on admissible mi-crostructures in the effective flow. We conclude that theconfiguration of the flow is effectively isotropic with an or-dering remnant.

Alexander PanchenkoWashington State Universitypanchenko@math.wsu.edu

MS11

Molecular Dynamics Models of Liquid CrystalElastomers

Liquid crystal elastomers are a fascinating class of mate-rials that combine the elasticity of rubber with the ne-matic or smectic order of liquid crystals. These materialshave been called “artificial muscles” because they rapidlychange shape under heating/cooling, applied fields, or op-tical illumination, with induced strains as high as 400%.We have developed a finite element elastodynamics tech-nique to model shape change in these materials, with ex-plicit coupling between liquid crystalline order and elasticstrain. We model several geometries, including a nematicelastomer artificial “earthworm” that moves via expan-sion/contraction combined with anisotropic friction. Oncevalidated, such models will serve as a bridge between fun-damental soft condensed matter theory and practical engi-neering design.

Robin SelingerLiquid Crystal Institute, Kent State Universityrobin@lci.kent.edu

MS12

Homogenization of Free Boundary Velocity

We study the homogenization limit of some (Hele-Shaw/Stefan type) free boundary problems with oscillatingfree boundary velocity. We first prove the existence of thelimiting free boundary velocity, which yields the uniformconvergence of the free boundaries in the homogenization

PD06 Abstracts 267

limit.

Inwon KimDepartment of MathematicsUCLAikim@math.ucla.edu

MS12

On the Contact-angle Condition for Stranski-Krastanow Islands

In this talk we present a rigorous derivation of the zerocontact-angle condition for Stranski-Krastanow islands instrained solid films.

Irene FonsecaCarnegie Mellon UniversityCenter for Nonlinear Analysisfonseca@cmu.edu

N. FuscoDipartimento di MatematicaUniversita di Napoli ”Federico II”nfusco@unina.it

M. MoriniSISSAmorini@sissa.it

Giovanni LeoniCarnegie Mellon Universitygiovanni@andrew.cmu.edu

MS12

Flame Propagation in Periodic Media

I will describe some results on the homogenization of afree boundary problem arising in the modeling of flamepropagation in premixed gas. The asymptotic regime cor-responds to a situation in which the flame width is smallerthan the characteristic length of the inhomogeneities of themedium.

Antoine MelletDepartment of MathematicsUniversity of Texas-Austinmellet@math.utexas.edu

Luis CaffarelliUniv of Texas at Austincaffarel@math.utexas.edu

Ki-Ahm LeeSeoul National Universitykiahm@math.snu.ac.kr

MS12

Self-propagating High tem-perature Synthesis (SHS) in the High ActivationEnergy Regime

We derive the precise limit of SHS in the high activation en-ergy scaling suggested by B.J. Matkowksy-G.I. Sivashinskyin 1978 and by A. Bayliss-B.J. Matkowksy-A.P. Aldushinin 2002. In the time-increasing case the limit turns outto be the Stefan problem for supercooled water with spa-tially inhomogeneous coefficients. Our precise form of thelimit problem suggest a strikingly simple explanation for

the numerically observed pulsating waves.

Regis MonneauCERMICS, Parismonneau@cermics.enpc.fr

Georg S. WeissGraduate School of Mathematical SciencesUniversity of Tokyogw@ms.u-tokyo.ac.jp

MS13

On Vortices in a Spinor Ginzburg–Landau Model

Recent papers in the physics literature have introducedspin-coupled (or spinor) Ginzburg–Landau models for com-plex vector-valued order parameters in order to accountfor ferromagnetic or antiferromagnetic effects in high-temperature superconductors and in optically confinedBose–Einstein condensates. These models give rise to newtypes of vortices, with fractional degree and non-trivial corestructure. By studying the associated system of equationsin R2 which describes the local structure of these vortices,we show some new and unconventional properties of thesevortices. We illustrate the various possibilites with somespecific examples of Dirichlet problems in the unit disk.These results are obtained in collaboration with L. Bron-sard, P. Mironescu, and E. Sandier.

Stanley AlamaMcMaster UniversityDepartment of Mathematics and Statisticsalama@mcmaster.ca

MS13

The Boundary Distribution of Surface Supercon-ductivity

It is well-known that when the applied magnetic field isin the range HC2 < hex < HC3 , superconductivity is con-centrated in a thin boundary layer. The distribution ofψ - the superconductivity order parameter- in this bound-ary layer is still unknown in some part of the above rangeof applied magnetic fields values. We prove, in the largeκ limit that when HC3 − εκ < hex the distribution of ψalong the boundary is uniform. This work is the result ofa collaboration with Bernard Helffer from Orsay.

Yaniv AlmogLousiana State UniversityDepartment of Mathematicsalmog@math.lsu.edu

MS13

Ginzburg-Landau Minimizers with PrescribedDegrees - Emergence of Vortices and Exis-tence/nonexistence of the Minimizers

Let Ω be a 2D domain with a hole ω. In the domainA = Ω\ω consider a class J of complex valued maps havingdegrees 1 and 1 on ∂Ω, ∂ω respectively. In a joint work withP. Mironescu we show that if cap(A) ≥ π (subcritical do-main), minimizers of the Ginzburg-Landau energy Eκ existfor each κ. They are vortexless and converge in H1(A) toa minimizing S1-valued harmonic map as the coherencylength κ−1 tends to 0. When cap(A) < π (supercriticaldomain), for large κ, we establish that the minimizing se-quences/minimizers develop exactly two vortices—a vortex

268 PD06 Abstracts

of degree 1 near ∂Ω and a vortex of degree −1 near ∂Ωwhich rapidly converge to ∂A. In this work it was conjec-tured that the global minimizers do not exist when κ > κ0.In a subsequent joint work with D. Golovaty and V. Ry-balko this conjecture was proved. The proof is based onan introduction of a related auxiliary linear problem whichallows for an explicit energy estimate.

Leonid BerlyandPenn State UniversityEberly College of Scienceberlyand@math.psu.edu

MS13

Vortices for a Rotating Toroidal Bose-EinsteinCondensate

We construct local minimizers of the Gross-Pitaevskiienergy, introduced to model Bose–Einstein condensates(BEC) which are subject to a uniform rotation. We con-sider the case where the condensate occupies a solid torusof revolution in R3 with starshaped cross-section. For largeenough angular speeds we produce local minimizers of theenergy which exhibit vortices, for small enough values ofthe length-scale parameter ε. These vortices concentrate atone or several planar arcs which minimize a line energy, ob-tained as a Γ-limit of the Gross-Pitaevskii functional. Thelocation of these limiting vortex lines can be described un-der certain geometrical hypotheses on the cross-sections ofthe torus. These results are obtained in collaboration withS. Alama and J.A. Montero.

Lia BronsardMcMaster Universitybronsard@mcmaster.ca

MS14

Fluid-Structure Interaction in Blood Flow

Fluid-structure interaction in blood flow will be discussed.

Suncica CanicDepartment of MathematicsUniversity of Houstoncanic@math.uh.edu

MS14

Stability of Compressible Vortex Sheets

Compressible vortex sheets are important in multidimen-sional compressible Euler flows. In this talk we will dis-cuss some recent developments in the theoretical analysisof compressible vortex sheets and related free boundaryproblems in multidimensional fluid dynamics and MHD.Some further trends and open problems on compressiblevortex sheets will also be addressed.

Gui-Qiang ChenNorthwestern UniversityUSAgqchen@math.northwestern.edu

MS14

Non-conservative Boltzmann Equations

We consider energy dissipative space homogeneous Boltz-mann Transport Equations (BTE) for interactions ofMaxwell type, which have a good representation in Fourier

space. Examples are inelastic BTE modeling granulargases or Elastic BTE for gas mixtures. We study self-similar solutions, usually referred as homogeneous coolingstates, their asymptotic behavior by analyzing their spec-tral properties in Fourier space. We show self-similar so-lutions with initial finite moments of all orders will havepower like tails, and so they can not have finite momentsfor all orders. This fact posses new challenges and issueson hydrodynamic fluid limit models. In addition, we showthe long time asymptotics is qualitatively similar to classi-cal non-linear wave equations. This work has been carriedout in collaboration with S. Bobylev and C. Cercignani.

Irene M. GambaDepartment of Mathematics and ICESUniversity of Texasgamba@math.utexas.edu

MS14

Traveling Waves in Thin Liquid Films Driven bySurfactant

In the lubrication approximation, the motion of a thin liq-uid film is described by a single fourth-order partial differ-ential equation that models the evolution of the height ofthe film. When the fluid is driven by a Marangoni forcegenerated by a distribution of insoluble surfactant, the thinfilm equation is coupled to an equation for the concentra-tion of surfactant. In this talk, I show the basic structure ofthis system, and begin an analysis of wave-like solutions inthe specific context of a thin film flowing down an inclinedplane. Numerical simulations reveal an array of travelingwaves, which persists when capillarity and surface diffusionare neglected. The analysis of the limiting system has somesurprises, and in this talk, I show how far we have come inunderstanding the numerical results analytically, and theanalytical results numerically.

Michael ShearerNorth Carolina State Univ.shearer@math.ncsu.edu

MS15

Control Problems in Fluid-Structure Interactions

Fluid structure interaction coupled system comprising ofa three dimensional Navier Stokes equation coupled to adynamic system of elasticity will be considered. The cou-pling takes place on the boundary – fixed interface betweenthe solid and the fluid. The following new results will bepresented: 1) wellposedness of weak and strong solutions;2) boundary feedback stabilization in the neighborhoods ofunstable equilibriums.

Irena LasieckaUniversity of Virginiail2v@virginia.edu

MS15

Control of Elastic Systems with Nonlinear Consti-tutive Law

We consider an elastic body in a region R0 ⊂ Rm, m = 2or 3. Our purpose in this talk is to explore the controlimplications of the use of nonlinear constitutive laws ob-tained from adding quartic terms to the quadratic potentialenergy expressions of the theory of linear elasticity. We in-dicate the effect of incorporating those terms and exploreconsequences for both static and dynamic control applica-

PD06 Abstracts 269

tions.

David RussellVirginia Techrussell@math.vt.edu

MS15

Mathematical Analysis of Aircraft Wing–AirflowInteraction

A model describing interaction of a long slender aircraftwing with subsonic inviscid isentropic potential air flowwill be presented. It is governed by a system of hyper-bolic PDEs with time convolution integral forcing terms.Asymptotic formulas for aeroelastic modes (wing vibra-tion frequencies) will be given. Expansion of a solution ofthe model equations with respect to the aeroelastic modeshapes will be presented. Application to flutter instabilityand its suppression will be discussed.

Marianna ShubovUniversity of New HampshireDepartment of Mathematics and Statisticsmarianna.shubov@euclid.unh.edu

MS15

On the Developments of Galin’s Stick-Slip ContactProblem

In this talk, an analytical solution procedure developedfor the the problem of indentation with friction of a rigidindenter into an elastic half-space is considered. The corre-sponding mixed boundary-value problem is formulated inplanar bipolar coordinates, and reduced to a singular inte-gral equation with respect to the unknown normal stress inthe slip zones. An exact analytical solution of this equationis constructed using the Wiener-Hopf technique, which al-lowed for a detailed analysis of the contact stresses, strain,displacement, and relative slip zone sizes. Also, a simpleanalytical solution is furnished in the limiting case of fullstick between the cylinder and half-space.

Olesya ZhupanskaUniversity of Floridazhupanska@reef.ufl.edu

MS16

Invariant Manifolds and the Stability of TravelingWaves in Burgers Equation

Invariant manifolds are constructed in the phase space ofperturbations to the traveling wave solution of Burgersequation and used to determine the asymptotic rate of con-vergence to the wave. This provides a geometric explana-tion of existing results demonstrating that the decay rateof perturbations can be increased by requiring that initialdata lie in appropriate algebraically weighted spaces.

Margaret BeckUniversity of SurreyDepartment of Mathematicsmabeck@math.bu.edu

C. Eugene WayneBoston UniversityDepartment of Mathematicscew@math.bu.edu

MS16

Evans Function Computation for Large Systems

We describe our new approach to Evans function compu-tation for large systems, which avoids the spatial and tem-poral blow up of exterior-product methods. We introducea polar-coordinate shooting method, for which the angu-lar equation is a variation of continuous orthogonalizationand the radial equation is easily computable and restoresanalyticity. We then use this new method to explore theshock wave stability problem for large one-dimensional vis-cous and viscous-dispersive conservation laws.

Jeffrey HumpherysBrigham Young Universityjeffh@math.byu.edu

Kevin ZumbrunIndiana UniversityUSAkzumbrun@indiana.edu

MS16

Multi-pulses in PDEs with Reflection and PhaseInvariance

We consider parameter-dependent dynamical systems withreflection and SO(2) symmetry which possess a homoclinicorbit to a saddle focus. The reflection symmetry is brokenby the wave speed. To address the existence of N-pulsesolutions and their PDE stability we use Lin’s method andLyapunov-Schmidt reduction. We discuss standing andtraveling 2 and 3-pulse solutions. Motivation comes fromthe complex cubic-quintic Ginzburg-Landau equation.

Bjorn SandstedeUniversity of Surreyb.sandstede@surrey.ac.uk

Vahagn E. ManukianNorth Carolina State Universityvemanuki@ncsu.edu

MS16

The Saddle-node of a Homogeneous Oscillation inSpatially Extended Reaction-diffusion Systems

We study the saddle-node bifurcation of a spatially homo-geneous oscillation in a reaction-diffusion system posed onthe real line. Beyond stability of the primary homogeneousoscillations created in the bifurcation, we investigate exis-tence, bifurcation and stability of wave trains with largewavelength that accompany the homogeneous oscillation.

Jens RademacherWeierstrass Institute for Applied Analysis and Stochasticsrademach@wias-berlin.de

Arnd ScheelSchool of Mathematics/IMAUniversity of Minnesotascheel@ima.umn.edu

MS17

On the Existence of Stokes Waves of Extreme Form

In this talk we present an alternative proof of the existenceof (weak) Stokes waves of extreme form in the finite depth

270 PD06 Abstracts

case. We also show the existence of a family of regularwaves. The approach is based on free boundary techniquesof Alt, Caffarelli, and Friedman.

Danut AramaCarnegie Mellon UniversityPittsburgh, PA 15213darama@andrew.cmu.edu

MS17

Some Problems in the Modeling and Analysis ofQuasi-static Evolution in Brittle Fracture

Some problems in the modeling and analysis of quasi-staticevolution in brittle fracture Based on Griffith’s criterionfor crack growth, a method was recently proposed for de-termining crack paths by taking continuous-time limits ofdiscrete-time variational problems. This has been success-fully carried out, but an important difference remains be-tween these solutions and Griffith’s model. I will explainthe method and the main issues in its implementation, andthen describe the remaining gap and some efforts to removeit.

Christopher LarsenWorcesterPolytechnic InstituteMath Deptcjlarsen@wpi.edu

MS17

Relaxed Energies for H1/2–Maps with Values intothe Circle

We consider, for maps in H1/2(R2;S1), a family of energiesrelated to seminorms equivalent to the standard one. Theseseminorms are associated to measurable matrix fields in thehalf space. For maps having finitely many prescribed topo-logical singularities, we show that the infimum of the en-ergy is equal to the lenght of a minimal connection relativeto a natural geodesic distance on the plane, connecting thesingularities. As a consequence, we compute the relaxedenergy functionals on smooth maps. This is a joint workwith A. Pisante.

Vincent MillotCarnegie Mellon UniversityPittsburgh, PA 15213vmillot@andrew.cmu.edu

MS17

Convergence of Equilibria of Thin Elastic Beams

In a series of recent papers a hierarchy of lower dimensionaltheories for nonlinearly elastic thin beams has been rigor-ously derived,starting from three-dimensional elasticity, bymeans of Gamma-convergence. This approach guaranteesconvergence of minimizers of the 3d elastic energy to min-imizers of the Gamma-limit problem.In this talk the con-vergence of (possibly non-minimizing) stationary points ofthe 3d elastic energy is discussed. These are joint workswith S. Muller and M. Schultz.

Maria Giovanna MoraSISSATrieste, Italymora@sissa.it

Stefan Muller, Maximilian SchultzMax-Planck Institute

Leipzig, Germanysm@mis.mpg.de, schultz@mis.mpg.de

MS18

A New Phase Space Based Formulation for Trans-mission Tomography

A new formulation is developed for travel time tomogra-phy. Our new method uses multiple arrival times and phaseinformation between the source and the measurement onthe boundary of the domain. The formulation can also dealwith anisotropic medium property. Numerical results showthat the unknown wave speed can be recovered with a goodaccuracy with fast convergence. This is a joint work withJianliang Qian, Gunther Uhlmann, and Hongkai Zhao.

Jianliang QianDepartment of MathematicsWichita State Universityqian@math.wichita.edu

Eric ChungUniversity of California IrvineDepartment of Mathematicstchung@math.uci.edu

Hongkai ZhaoUniversity of California, IrvineDepartment of Mathematicszhao@math.uci.edu

Gunther UhlmannUniversity of Washingtongunther@math.washington.edu

MS18

A Lax-Friedrichs Sweeping Method for StaticHamilton-Jacobi Equations

In this talk, we will present a Lax-Friedrichs fast sweep-ing method to approximate the viscosity solution of staticHamilton-Jacobi equations. By solving for the value of aspecific grid point in terms of its neighbors and incorporat-ing a group-wise causality principle into the Gauss-Seideliteration, we have an easy-to-implement and fast conver-gent numerical method. For computational boundary con-dition, we give a simple recipe which enforces a version ofdiscrete min-max principle. Some convergence analysis isdone for the one-dimensional eikonal equation. Extensive2-D and 3-D numerical examples illustrate the efficiencyand accuracy of the approach.

Chiu-Yen KaoUniversity of MinnesotaInstitute for Mathematics and its Applicationskao@ima.umn.edu

MS18

On the Complexity of Fast Sweeping Algorithms

We explore—both theoretically and empirically—the com-putational complexity of fast sweeping algorithms. Whilethere are no surprises in two dimensions, the situation forthree dimensions is less encouraging than it might first ap-pear. On the empirical side, we also present a flexiblenew code base for approximating the solution of stationaryHamilton-Jacobi equations, and discuss some of the trade-

PD06 Abstracts 271

offs in the choice of Python as an implementation language.

Ian M. MitchellUniversity of British ColumbiaDepartment of Computer Sciencemitchell@cs.ubc.ca

MS18

Fast Sweeping Methods for Eikonal Equations onTriangular Meshes

The original fast sweeping method relies on natural order-ing provided by a rectangular mesh. We propose novel or-dering strategies so that the fast sweeping method can beextended efficiently and easily to any unstructured mesh.To that end we introduce multiple reference points andorder all the nodes according to their distances to thosereference points. We show extensive numerical examplesto demonstrate the accuracy and efficiency of the new al-gorithm on triangular meshes.

Hong-Kai ZhaoUniversity of California, Irvinezhao@math.uci.edu

Jianliang QianDepartment of MathematicsWichita State Universityqian@math.wichita.edu

Yong-Tao ZhangUniversity of California, Irvinezyt@math.uci.edu

MS19

Nonlinear Fourth-order Parabolic Equation andRelated Entropy Functionals

A nonlinear fourth-order parabolic equation is presented;the equation appears in quantum semiconductor modelingand also in the study of interface fluctuation in spin system.Several Lyapunov functionals are used in order to show ex-istence results and long-time asymptotics for bounded andunbounded domains in one and more dimensional space.Exponential decay in time of solutions to the steady stateis proved by the entropy-entropy production method usinglogarithmic Sobolev inequalities.

Maria GualdaniThe University of Texas at Austingualdani@ices.utexas.edu

MS19

A Semiclassical Model for Thin Quantum Barriers

We present a time-dependent semiclassical transport modelfor mixed state scattering with thin quantum barriers. Theidea is to use a multiscale approach as a means of connect-ing regions for which a classical description of the systemdynamics is valid across regions for which the classical de-scription fails, such as when the gradient of the poten-tial is undefined. We do this by first solving a stationarySchrodinger equation in the quantum region to obtain thescattering coefficients. These coefficients allows us to buildthe interface condition to the particle flux that bridgesthe quantum region, connecting the two classical domains.Away from the barrier, the problem may be solved by tradi-tional numerical methods. Therefore, the overall numerical

cost is roughly the same as solving a classical barrier. In theone-dimensional case, we use a finite-volume method thatextends the Hamiltonian-preserving scheme introduced byJin and Wen for a classical barrier. In the two-dimensionalcase, we consider a mesh-free particle method that can becomputed efficiently and that may be extended to higher-dimensions.

Shi Jin, Kyle NovakUniversity of Wisconsin at Madisonjin@math.wisc.edu, novak@math.wisc.edu

MS19

Simulating the Wigner Equation with the Discon-tinuous Galerkin Method and a Non-polynomialApproximation Space

The Wigner equation, with a suitable collision operator,models the interaction between a quantum system and itsenvironment. It is a useful model for semiconductor de-vices, and current simulations are typically based on oper-ator splitting methods. We demonstrate simulation via theDiscontinuous Galerkin method, using an oscillatory basis.The use of a non-polynomial approximation space for DGcalculations has recently been proposed by Yuan and Shu,and promises greater accuracy on a coarser mesh.

Richard SharpUniversity of Texas at Austinrsharp@ices.utexas.edu

MS20

Optical Manipulation of Matter Waves

See MS16 in under annual meeting (AN06 abstracts.)

Ricardo Carretero-GonzalezDepartment of MathematicsSan Diego State Universitycarreter@sciences.sdsu.edu

MS20

Soliton Dynamics and Scattering for the Gross-Pitaevskii Equation

See MS16 in under annual meeting (AN06 abstracts.)

Stephen GustafsonDepartment of MathematicsUniversity of British Columbiagustaf@math.ubc.ca

MS20

Topological Effects with Matter Waves: A NewTwist on BEC

See MS16 in under annual meeting (AN06 abstracts.)

Chandra RamanPhysics DepartmentGeorgia Tech.chandra.raman@physics.gatech.edu

MS20

High Temperature Superfluidity and the Super-fluid - Mott Insulator Transition in Ultracold

272 PD06 Abstracts

Atomic Gases

See MS16 in under annual meeting (AN06 abstracts.)

Christian SchunckMITsch

MS21

Exact Solutions to Supersonic Flow onto a SolidWedge

We present a problem arising in supersonic flow onto a solidwedge and show how to construct exact solutions for thecompressible potential flow model.

Tai-Ping LiuStanford UniversityTaiwan/USAliu@math.stanford.edu

Volker W. EllingBrown Universityvolker@dam.brown.edu

MS21

Traffic Congestion-An Instability in a HyperbolicSystem

In this talk I’ll discuss second order traffic models and showthey support stable oscillatory traveling waves typical ofthose seen on a congested roadway. These solutions arisewhen the trivial solution of uniformly spaced cars travelingat constant velocity is unstable. Wave profiles and speedswill be exhibited.

James M. GreenbergCarnegie Mellon UniversityDept of Mathematical Sciencesgreenber@andrew.cmu.edu

MS21

Hyperbolic Conservation Laws on Manifolds

The theoretical work on discontinuous solutions of nonlin-ear hyperbolic conservation laws in several space variableshas been restricted so far to problems set in the Euclid-ian space. Motivated by numerous applications, to geo-physical fluid flows and general relativity in particular, Iwill attempt to set the foundations for a study of entropysolutions to hyperbolic conservation laws on a manifold.Understanding the interplay between the geometry of themanifold and the behavior of discontinuous solutions ofpartial differential equations is particularly challenging. Inthis lecture, I will discuss the derivation of the L1 contrac-tion property and the total variation diminishing property,which are known to play a central role in the theory andthe numerical analysis of hyperbolic problems. Next, I willpresent the generalization to manifolds of, both, Kruzkov’sand DiPerna’s well-posedness theories. Finally, I will es-tablish the convergence of the finite volume schemes onmanifolds, with a special attention for the case of the 2-sphere. This a joint work with M. Ben-Artzi (Jerusalem).

Philippe G. LeFlochUniversity of Paris 6Laboratoire Jacques-Louis Lionslefloch@ann.jussieu.fr

MS21

Computing Multi-valued Solutions for Euler-Poisson Equations

In this talk we first review the critical threshold phenom-ena for Euler-Poisson equations, which arise in the semi-classical approximation of Schrodinger-Poisson equationsand plasma dynamics. We then present a novel levelset method for the computation of multi-valued velocityand electric fields of one-dimensional Euler-Poisson equa-tions. This method uses an implicit Eulerian formulationin an extended space—called field space, which incorpo-rates both velocity and electric fields into the configura-tion space. Multi-valued velocity and electric fields arecaptured through common zeros of two level set functions,which solve a linear homogeneous transport equation in thefield space. The evaluation of density is also discussed.

Zhongming WangIowa State Universityericwang@iastate.edu

Hailiang LiuIowa State UniversityUSAhliu@iastate.edu

MS22

Homogenization of Integral Energies via the Peri-odic Unfolding Method

We consider the periodic homogenization of nonlinear in-tegral energies of the type

(1) u ∈W 1,p(Ω;Rm) →∫

Ω

f(x

εh,∇u

)dx,

where Ω is a bounded open subset of Rn with a Lipschitzboundary, and f is a Caratheodory energy density satisfy-ing{

f : (x, z) ∈ Rn × Rnm → f(x, z) ∈ [0,+∞[,f( · , z) Lebesgue measurable, Y -periodic for everyz ∈ Rnm, f(x, · ) continuous a.e. x ∈ Rn.

(4)with Y the reference cell ]0, 1[n. Assume that f(x, ·) isquasiconvex for a.e. x ∈ Rn, and furthermore that, forp ∈ [1,+∞[, M > 0, and an Y -periodic a ∈ L1(Y ), itsatisfies the following growth conditions:{

f(x, z) ≤ a(x) +M |z|p for a.e. x ∈ Rn, and every z ∈ Rnm,vertz|p ≤ f(x, z) for a.e. x ∈ Rn, and every z ∈ Rnm.

(5)It is then known that as εh → 0, one has

inf{

lim infh→+∞∫Ωf(

xεh,∇uh

)dx :

{uh} ⊂W 1,p(Ω;Rm), uh → u in Lp(Ω;Rm)}

= inf

{lim suph→+∞

∫Ωf(

xεh,∇uh

)dx :

{uh} ⊂W 1,p(Ω;Rm), uh → u in Lp(Ω;Rm)

}=∫Ωfhom(∇u)dx,

where the homogenized energy density fhom is defined by

fhom : z ∈ Rnm → limt→+∞

1

tninf

{∫tY

f(y, z + ∇v)dy :

PD06 Abstracts 273

v ∈W 1,p0 (tY ;Rm)

}.

This convergence result was established in [1] and [2] bysophisticated Γ-convergence arguments. Since then, manyattempts at simplifying the proof (for example by usingtwo-scale convergence), seem not to have borne fruit. In[5], a joint paper with Alain Damlamian and Riccardo DeArcangelis, we apply the tool of periodic unfolding from [3].This gives a direct proof of the convergence result (underslightly weaker assumptions than in [1] and [2]), making useof simple weak convergence arguments in Lp-type spaces.This paper is part of a series of ongoing works concerningthe applications of the periodic unfolding method to ho-mogenization. The first one [4], treated the same problembut in the case of convex densities.

Doina CioranescuUniversite Pierre et Marie Curiecioran@ann.jussieu.fr

MS22

The Unfolding Approach for Periodic Homogeniza-tion of Non Linear Elliptic PDE’s and for theMosco-convergence of Convex Integrals

In this presentation, we show how the use of the periodicunfolding method greatly simplifies and clarifies the ho-mogenization of non-linear elliptic operators (in the mul-tivalued framework generalizing the Leray-Lions type, asa map from the space W 1,p

0 (Ω) to its dual space). In thecase of subdifferentials of convex functions, the result canbe expressed in terms of the Mosco-convergence of the cor-responding sequence of integral functionals on the spaceW 1,p

0 (Ω).

Alain DamlamianUniversite Paris 12 Val de Marne94010 Creteil Cedex, Francedamla@univ-paris12.fr

MS22

Asymptotics of a Spectral Problem Associated withthe Neumann Sieve

We analyze the asymptotic behavior of a Steklov spectralproblem associated to the Neumann Sieve model, i.e a threedimensional set Ω, cut by a hyperplan Σ where each of thetwo-dimensional holes, ε− periodically distributed on Σ,have diameter rε. Depending on the asymptotic behaviorof the ratios rε

ε2we find the limit problem of the ε spectral

problem and prove that the sequences λεn, formed by the

n-th eigenvalue of the ε problem, converge to λn, the n-th eigenvalue of the limit problem, for any n ∈ N. Wealso prove the weak convergence, on a subsequence, of theassociated sequence of eigenvectors uε

n, to an eigenvectorassociated to λn. When λn is a simple eigenvalue, we showthat the entire sequence of the eigenvectors converges. Asa consequence, similar results hold for the spectrum of theDtN map associated to this model.

Daniel OnofreiDept. Mathematical SciencesWorcester Polytechnic Instituteonofrei@wpi.edu

Bogdan M. VernescuWorcester Polytechnic InstDept of Mathematical Sciences

vernescu@wpi.edu

MS23

Weak and Strong Global Attractors for the 3DNavier-Stokes Equations

Often the evolution of an autonomous dissipative dynam-ical system can be described by a semigroup of solutionoperators. If this semigroup is asymptotically compact,then the classical theory of attractors yields the existenceof a compact global attractor. However, for some dynami-cal systems the semigroup is not asymptotically compact.Some dynamical systems do not even possess the semigroupdue to the lack of uniqueness (or proof of uniqueness). Wewill define and study such objects as an omega-limit, in-variant set, and global attractor for a dynamical systemthat is not well-posed. Then we will apply the obtainedresults to the 3D incompressible Navier-Stokes equations.

Alexey CheskidovDepartment of MathematicsUniversity of Michiganacheskid@umich.edu

MS23

The Weak Attractor of the 3-D Navier-StokesEquations

We study the pojection of the weak attractor of the 3Dperiodic Navier-Stokes in the normalized, dimensionlessenergy-enstrophy plane, obtaining restrictive bounds onthe attractor’s location. These bounds imply that the con-ditions necessary for the direct energy cascade can occureither near the origin of the phase space or near a singu-larity.

Ciprian FoiasTexas A&M Universityfioas@math.tamu.edu

Michael S. Jolly, Radu DascaliucIndiana UniversityDepartment of Mathematicsmsjolly@indiana.edu, rdascali@indiana.edu

MS23

The Attractor of the 2-D Navier-Stokes Equationsin the Energy-enstrophy Plane

We examine how the global attractor of the 2-D Navier-Stokes equations projects in the normalized, dimensionlessenergy-enstrophy plane. A particular motivation is to seewhether a condition for a direct energy cascade can beachieved. For any force, the attractor must lie betweena line (the Poincare inequality) and a parabola. An op-timization of the bounding parabola results in a sharperbound, yet preliminary computations suggest that an evensharper, linear estimate holds.

Ciprian FoiasTexas A & M,Indiana Universityfoias@math.tamu.edu

Radu DascaliuDepartment of MathematicsIndiana Universityrdascali@indiana.edu

274 PD06 Abstracts

Michael S. JollyIndiana UniversityDepartment of Mathematicsmsjolly@indiana.edu

MS23

Forced Two Layer Beta-plane Quasi-geostrophicFlow

We consider a model of quasigeostrophic turbulence thathas proven useful in theoretical studies of large scale heattransport and coherent structure formation in planetaryatmospheres and oceans. The model consists of a cou-pled pair of hyperbolic PDE’s with a forcing which repre-sents domain-scale thermal energy source. Although theuse to which the model is typically put involves gatheringinformation from very long numerical integrations, littleof a rigorous nature is known about long-time propertiesof solutions to the equations. First we define a notion ofweak solution, and show using Galerkin methods the long-time existence and uniqueness of such solutions. Then weshow that the unique weak solution produces, via the in-verse Fourier transform, a classical solution for the system.Moreover, we prove that this solution is analytic in spaceand positive time.

Lee PanettaTexas A&M Universitypanetta@ariel.met.tamu.edu

Constantin OnicaIndiana UniversityInstitute for Sci. Comp. and Appl. Math.conica@indiana.edu

MS24

Optimal Structures of Multiphase Composites andtheir Fields

The translation bounds for multiphase mixtures are sharponly in a range of volume fractions of components. Weanalyse the fields in translation-optimal structures for two-and three-dimensional conductive composites and describethe largest possible classes of multimaterial structures thatachieve the bounds as well as the limits of attainability ofthese bounds. The two schemes of synthesis of optimalstructures are suggested. One is an ”inverse laminationscheme” in which the rank-one connected fields in lami-nates are fixed but the geometry is varied, the second isan adjusted differential scheme. Our results bring togetherand expand classes of optimal microgeometries found ear-lier by Milton, Milton and Kohn, Lurie and Cherkaev,Gibiansky and Cherkaev, Gibiansky and Sigmund. Wealso discuss new supplementary bounds for multicompo-nent mixtures and structures that realize these bounds.Some results were obtained in collaboration with VincenzoNezi.

Andrej V. CherkaevDepartment of MathematicsUniversity of Utahcherk@math.utah.edu

Nathan AlbinUniversity of Utahna

MS24

Solute Transport in Porous Media with NetworkMicrogeometry

We study solute transport in porous media with periodicmicrostructures consisting of collections of interconnectedby thin channels. We identify a physical mechanisms thatpromotes mixing of the solute with the host liquid and ob-tain the macroscopic transport equation that the concen-tration of solute satisfies. We discuss examples motivatedby the transport of nutrients in bones.

Guillermo GoldszteinGeorgia TechAtlanta, GAggold@math.gatech.edu

MS24

Optimal Lower Bounds on the L∞ Norm of theStress in Random Elastic Composites

Consider a cube containing two isotropic linear elastic ma-terials. The only information given on the configurationof the two materials is the measure of the set occupied byeach material. Subject to this limited geometric informa-tion one is interested in the L∞ norm of a specified stressinvariant when the domain is subject to a uniform forceon the boundary. We provide a methodology for derivingoptimal lower bounds on the L∞ norm for various stressinvariants when only the volume fraction of each materialis known.

Robert P. LiptonDepartment of MathematicsLouisiana State Universitylipton@math.lsu.edu

MS24

Fluctuating Fractal Surfaces

We construct a family of fractal surfaces with fluctuatinggeometry at all scales and describe some metric and an-alytic properties of these sets from the point of view ofcomposite media.

Umberto MoscoDepartment of Mathematical SciencesWorcester Polytechnic Institutemosco@WPI.EDU

MS25

Shocks in Driven Liquid Films

Driven contact line problems in thin liquid films are an ac-tive area of research. The mathematical theory of shockwaves has recently been shown to play an important rolein our understanding of basic properties of the contact linemotion. I will present the theory for two recently studiedexperimental systems: (1) Thermally driven films coun-terbalanced by gravity are described by a scalar conserva-tion with a non-convex flux. Such systems are known toproduce ‘undercompressive shocks’ in which characteristicsemerge from the shock on one side. (2) A related problemis that of particle laden flow driven by gravity. The dif-ferential settling rate of the particles with respect to thefluid results in the formation of double shock fronts whichare solutions of a system of two conservation laws for themotion of the species. Comparison between theory and ex-periment will be discussed, along with open mathematical

PD06 Abstracts 275

problems directly related to the experiments.

Andrea L. BertozziUCLA Department of Mathematicsbertozzi@math.ucla.edu

MS25

The Hele-Shaw Problem as a ”Mesa” Limit of Ste-fan Problems: Existence, Uniqueness, and Regu-larity of the Free Boundary

We study a Hele-Shaw problem as a ”Mesa” type limit ofone phase Stefan problems in exterior domains. We studythe convergence, determine some of qualitative propertiesof the unique limiting solution, and prove regularity of thefree boundary of this limit under very general conditionson the initial data. Indeed our results handle changes intopology and multiple injection slots. This project is jointwork with Marianne Korten and Charles Moore.

C. N. MooreDepartment of MathematiscKansas State Universitycnmoore@math.ksu.edu

Ivan BlankDepartment of MathematicsWorcester Polytechnic Instituteblanki@WPI.EDU

Marianne KortenDepartment of MathematicsKansas State Universitymarianne@math.ksu.edu

MS25

Free Boundary Measures and Partial Regularity ofSolutions in a Mixed Type Evolution Equation

Solutions of the mixed type equation

ut = Δxα(u) + divxF (u),

where x ∈ IRn, α(u) is a continuous piecewise linear func-tion such that α(u) = 0 for |u| ≤ 1 and having slope 1 oth-erwise, and F (u) is a Lipschitz from IR to IRn, combine thefeatures of conservation laws, and of free boundary prob-lems. We will discuss the regularity of weak solutions tothis equation, in particular, we will show that for solutionslocally in L2, α(u) satisfies local energy estimates, and thatu is continuous in the set {|u| > 1}. With this in hand wewill identify the free boundary measures supported on theboundary of {|u| > 1}. This allows us to study separatelythe jumps in u at the free boundary and the discontinu-ities occuring within the region {|u| < 1} governed by theconsevation law.

G.-Q. ChenDepartment of MathematicsNorthwestern Universitygqchen@math.northwestern.edu

C. N. MooreDepartment of MathematiscKansas State Universitycnmoore@math.ksu.edu

Monica TorresDepartment of Mathematics

Purdue Universitytorres@math.purdue.edu

Marianne KortenDepartment of MathematicsKansas State Universitymarianne@math.ksu.edu

MS25

A Deterministic Control Based Approach to Cur-vature Flows

In a joint work with Bob Kohn, we give a new control-typeinterpretation on the level-set approach to motion by cur-vature and related interface motion laws. More precisely,we give a family of discrete-time, two-person games whosevalue functions converge in the continuous-time limit to thesolution of the motion-by-curvature PDE. The value func-tion of a deterministic control problem is normally a first-order Hamilton-Jacobi equation, while the level-set formu-lation of motion by curvature is a second-order parabolicequation.

Robert V. KohnCourant InstituteNew York Universitykohn@cims.nyu.edu

Sylvia SerfatyCourant Institute, NYUNew York, NYserfaty@cims.nyu.edu

MS26

Lawrence-Doniach System for High-temperatureSuperconductors

We consider the Lawrence-Doniach model for layered su-perconductors, which describes a large class of high-temperature superconductors as a finite number of parallelsuperconductors, with Josephson coupling between them,including nonlinear Maxwell’s equations that describe theinduced magnetic field. Assuming an applied magneticfield which is nontangential to the layers, we prove thata change of phase from superconducting to normal occurs,and we estimate the upper critical field in terms of the ma-terial parameters, the Josephson constant, and the angleof the nontangential applied field. Our estimates extendto the nonlinear regime predictions by physicists based onthe linearized system.

Patricia BaumanPurdue Universitybauman@math.purdue.edu

Yangsuk KoUniversity of California, Bakersfieldyangsuk ko@firstclass1.csubak.edu

MS26

Bifurcation Structure of a Ginzburg-Landau Modelin a Ring

We deal with the one-dimensional Ginzburg-Landau energyfunctional which is a simplified model of the superconduc-tivity in a thin tubular ring. The Euler-Lagrange equationof this functional, called the Ginzburg-Landau equation,has two physical parameters. We show a global bifurcation

276 PD06 Abstracts

structure in the parameter space if the tubular domain hasuniform thickness. Moreover we exhibit a local bifurcationdiagram around a singularity for the case of nonuniformthickness.

Yoshihisa MoritaRyukoku UniversityDepartment of Appl. Math. and Informaticsmorita@rins.ryukoku.ac.jp

MS26

Transitions of Liquid Crystals and Critical Valuesof Elastic Coefficiants

We use the Landau-de Gennes model to examine the effectsof twist and bend to liquid crystals. In the case of equalelastic coefficients (Ki = K2 = K3 = K) we show thatthere exists a critical value Kc such that the liquid crystalis in the smectic state if K < Kc and is in the nematic stateif K > Kc. Under some non-degeneracy and regularityassumptions on the nematic states we find the precise valueof Kc and obtain the asymptotic behavior of minimizers ofEK as K approach Kc.

Xingbin PanEast China Normal Universityxbpan@euler.math.ecnu.edu.cn

MS26

Something About Ginzburg-Landau Vortices in theHigh-Kappa Limit

I will talk either about vortex collisions in the Ginzburg-Landau heat flow or about some refined Gamma-convergence results for Ginzburg-Landau with appliedmagnetic field

Sylvia SerfatyCourant Institute, NYUNew York, NYserfaty@cims.nyu.edu

MS27

Dependence of Large Entropy Solutions to the Eu-ler Equations on Physical Parameters

We compare the entropy solution Uμ to a hyperbolic sys-tem of conservation laws for which the flux function de-pends on a parameter vector μ = (μ1, . . . , μk) with theStandard Riemann Semigroup S generated when μ ≡ 0and establish L1 error estimates pointwise in time with re-spect to the flux parameter μ. We apply our results tothe isentropic and relativistic Euler equations with largeoscillations for which the parameters are the adiabatic ex-ponent γ > 1 and the speed of light c < ∞ in comparisonwith the isothermal Euler equations.

Yongqian ZhangFudan University–

Gui-Qiang ChenNorthwestern UniversityUSAgqchen@math.northwestern.edu

Cleopatra ChristoforouNorthwestern University

Department of Mathematicscleo@math.northwestern.edu

MS27

A Particular Large Data Solution of the 1-D EulerSystem

It is known from explicit examples of finite time blowupin BV and L∞ that the class of strictly hyperbolic sys-tems is too large to allow a general global existence resultfor large data. On the other hand it is hoped that physi-cal systems, and especially the Euler system, have enoughstructure to admit global weak solutions, even for largeamplitude/variation data. We present an example of a so-lution to the Euler system which exhibits the same typeof interaction pattern for which blowup has been shown inother systems. The data giving rise to these particular so-lutions may be arbitrarily large in BV or L∞. By carefullyestimating the interactions we show that no blowup occursfor these particular solutions.

Erik EndresPennsylvania State UniversityDepartment of Mathematicseee121@psu.edu

Kris JenssenPennsylvania State Universityhkj1@psu.edu

MS27

Self-similar Solutions for Transonic 2D RiemannProblems

We discuss recent developments of transonic self-similar 2DRiemann problems.

Eun Heui KimCalifornia State University at Long Beachekim4@csulb.edu

MS27

Transonic Shock Solutions for a System of Euler-Poisson Equations , 1-d Case

A boundary value problem for a 1-d system of Euler-Poisson equations is considered. The boundary conditionsare supersonic on the left end and subsonic on the rightend. The solutions with transonic shocks are constructed,and the shock locations are shown to be uniquely and ex-plicitly determined by the boundary data.

Tao LuoGeorgetown Universitytl48@georgetown.edu

MS28

Artificial Compressibility Method for the Incom-pressible Navier Stokes Equations

We study an hyperbolic approximation of the Leray solu-tion to the 3D Navier Stokes equation. In particular we de-scribe an hyperbolic version of the so called compressibiltymethod investigated by J.L.Lions, Temam. This approx-imation is motivated by numerical analysis applicationswhere, in order to overcome the difficulty related the di-vergence free condition, an artificial compressibilty is intro-duced. We will study the problem on unbounded domains

PD06 Abstracts 277

by replacing Sobolev compact embedding with the use ofdispersive estimates of Strichartz type combined with bi-linear estimates.

Donatella DonatelliUniversity of L’Aquila -Italydonatell@univaq.it

Pierangelo MarcatiDipartimento di Matematica, Universita di L’Aquiladonatell@univaq.it

MS28

The Existence of Weak, Small Energy Solutions forthe Equations of Motion of 3D Compressible, Vis-cous Fluid Flows with the No-slip Boundary Con-ditions

We consider the equations of motion of a compressible, vis-cous, isentropic fluid in a bounded domain in R2 or R3 withthe no-slip boundary conditions. Given a constant, equi-librium state we construct a global in time, regular weaksolution, provided that initial data are close to the quilib-rium when measured by weak norms and discontinuities inthe initial density decay near the boundary.

Mikhail PerepelitsaUniversity of Massachusettsmikhailp@math.nothwestern.edu

MS28

Boundary Layer Analysis for Isentropic Gas Dy-namics with Real Viscosity and Related Systems

We study the problem of boundary layer formation for vis-cosity approximations of the isentropic gas dynamics, in-cluding source terms as friction and topography, for phys-ical viscosity models. The case of Saint-Venant equationsfor shallow water is also discussed.

Chiara SimeoniUniversite de Nice - Sophia AntipolisLaboratoire J. .A. Dieudonnesimeoni@math.unice.fr

Debora AmadoriUniversity of L’Aquila (Italy)amadori@univaq.it

MS28

Homogenization in Parabolic Equations

We are interested in homogenization of parabolic equa-tions including diffusion equation, Incompressible Navier-Stokes equation in reticulated domain or perforated do-main, which are arose from cell biology, rheology. In thelimit, we identify the effective parabolic equations.

Bo SuIowa State University - USAbosu@iastate.edu

MS29

On Singular Limits for the Full Navier-Stokes-Fourier System

We discuss some singular limits of global-in-time solutionsto the full Navier-Stokes-Fourier system describing the dy-

namics of a compressible, viscous, and heat conductingfluid. In particular, we concentrate on the case when theMach number tends to zero. We shall show that in thelimit we arrive at the Oberbeck-Boussinesq approximation.Several generalizations of this result are also discussed, inparticular the case of a chemically reacting fluid.

Eduard FeireislMathematical Institute ASCR, Zitna 25, 115 67 Praha 1Czech Republicfeireisl@math.cas.cz

MS29

Lagrangean Structure and Propagation of Singu-larities in Multidimensional Compressible Flow

Not available at time of publication.

David HoffIndiana Universityhoff@indiana.edu

Marcelo dos SantosUniversity of Campinas, Brazilmsantos@ime.unicamp.br

MS29

Navier’s Slip and Incompressible Fluids with Tem-perature, Pressure and Shear Rate Dependent Vis-cosity

There is compelling experimental evidence for the viscos-ity of a fluid depending on the temperature, the shear rateas well as the mean normal stress (pressure). Moreover,while the viscosity can vary by several orders of magnitudethe density suffers very minor variation, when the rangeof the pressure is sufficiently large, thereby providing jus-tification for considering the fluid as being incompressiblewhile at the same time possessing a viscosity that is de-pendent on the pressure. We investigate the mathematicalproperties of internal unsteady three-dimensional flows ofsuch fluids subject to Navier’s slip at the boundary. We es-tablish the long-time existence of a weak solution for largedata provided that the viscosity depends on the tempera-ture, the shear rate and the pressure in a suitably specifiedmanner. This specific relationship however includes theclassical Navier-Stokes-Fourier fluids and power-law fluidsas special cases. This talk is based on joint work with M.Bulicek and K.R. Rajagopal.

Josef MalekCharles University, PragueMathematical Institutemalek@karlin.mff.cuni.cz

MS30

The Control-theoretic Approach to Construction ofFast Iterative Solvers

We outline the dynamical systems/optimal control per-spective on the design of numerical approximations. Inparticular, we describe a natural class of process level dis-cretizations of a continuous time/continuous state deter-ministic optimal control problem, when the goal is thatthe related iterative solver for the dynamic programmingequation should converge with a number of iterations thatis independent of the discretization. An extension to de-

278 PD06 Abstracts

terministic differential games is also described.

Paul DupuisBrown UniversityDivision of Applied Mathematicsdupuis@cfm.brown.edu

MS30

An Application of Fast Sweeping Methods toTransmission Traveltime Tomography

Traditional transmission travel-time tomography hinges onray tracing methods. We propose a new PDE-based ap-proach which avoids using the cumbersome ray-tracingtechnique. We start from the eikonal equation, define amismatching functional and then derive the gradient ofthe functional using an adjoint state method. These com-putations can be efficiently done using the fast sweepingmethod. We demonstrate the robustness of the methodwith examples including the Marmousi synthetic velocitymodel.

Shingyu LeungUniversity of California Los AngelesDepartment of Mathematicssyleung@math.ucla.edu

Jianliang QianWichita State UniversityDepartment of Mathematicsqian@math.wichita.edu

MS30

A Fast Sweeping Method Based on Discontinu-ous Galerkin Methods for Static Hamilton-JacobiEquations

We will report on our preliminary results on combin-ing the discontinuous Galerkin methods, a properly cho-sen numerical Hamiltonian, and Gauss-Seidel iterationswith alternating-direction sweepings to design a high or-der fast sweeping method for static Hamilton-Jacobi equa-tions. The main new feature of the method is to explorethe advantages of the local compactness of the discontinu-ous Galerkin methods which fits well in the fast sweepingframework. A description of the algorithm and preliminarynumerical results for Eikonal equations will be presented.

Hong-Kai ZhaoUniversity of California, Irvinezhao@math.uci.edu

Fengyan LiUniversity of South Carolinafli@math.sc.edu

Chi-Wang ShuDivision of Applied MathematicsBrown Universityshu@dam.brown.edu

Yong-Tao ZhangUniversity of California, Irvinezyt@math.uci.edu

MS30

Wide Stencil Schemes for First and Second Order

Degenerate Elliptic Equations

We will present a useful class of wide stencil schemes fordegenerate elliptic equations. These types of schemes havebeen used to build convergent schemes for several typesof second order equation, including: Level Set Motion byMean Curvature, the Monge-Ampere equation, the InfinityLaplacian, the equation for the convex envelope. Theseschemes are easy to build, and are provably convergent.We will discuss adapting these schemes to build schemesfor Hamilton-Jacobi equations on unstructured grids.

Adam ObermanSimon Fraser UniversityDepartment of Mathematicsoberman@math.sfu.ca

MS31

Existence and Stability of Traveling Waves of theRegularized Short Pulse and Ostrovsky Equations

We consider the question of existence and stability of trav-eling waves for the regularized short pulse and Ostrovskyequations. The regularization term in the short pulse equa-tion arises naturally as a higher order expansion term inthe derivation of the original short pulse equation. The ex-istence of single bump traveling waves for both equationsis proved via the Fenichel theory for singularly perturbedODEs and a Melnikov type transversality calculation. Wethen consider the spectral stability of the waves via topo-logical arguments based on the augmented Evans bundle,which is constructed from the fast-slow decomposition ofthe underlying traveling wave.

Christopher JonesUniversity of North CarolinaDepartment of Mathematicsckrtj@email.unc.edu

Nicola D. CostanzinoBrown UniversityUNC-Chapel Hillncostanz@email.unc.edu

MS31

On Subsonic Detonation Waves in Inert PorousMedium

We consider Sivashinsky’s model of subsonic detonationthat describes propagation of combustion fronts in highlyresistable media. It is known that there exists a travel-ing wave asymptotically connecting the unburnt and burntstates, which is unique if thermal diffusivity is neglected.Using geometric singular perturbation theory we study theuniqueness and stability of the wave in the case of positivethermal diffusivity.

Anna GhazaryanUniversity of North Carolina at Chapel Hillghazaryan@niss.org

Chris JonesDepartment of MathematicsUniversity of North Carolinackrtj@email.unc.edu

Peter GordonNew Jersey Institute of Technologypeterg@oak.njit.edu

PD06 Abstracts 279

MS31

Rigorous Asymptotic Expansions for Critical WaveSpeeds in a Family of Scalar Reaction-diffusionEquations

We investigate traveling waves in the family of reaction-diffusion equations given by

ut = uxx − 2um(1 − u).

For m ≥ 1 real, there is a critical wave speed ccrit(m)which separates waves of exponential structure from thosethat decay only algebraically. We derive rigorous asymp-totic expansions for ccrit by perturbing off two solv-able cases, the classical Fisher-Kolmogorov-Petrowskii-Piscounov equation (m = 1) and a degenerate cubic equa-tion (m = 2). In addition, we study the asymptotic criticalspeed in the limit as m→ ∞. Our approach uses geometricsingular perturbation theory, as well as the blow-up tech-nique, and confirms results previously obtained throughasymptotic analysis.

Tasso J. KaperBoston UniversityDepartment of Mathematicstasso@math.bu.edu

Freddy DumortierUniversity of Hasseltfreddy.dumortier@uhasselt.be

Nikola PopovicBoston UniversityCenter for BioDynamics and Department of Mathematicspopovic@math.bu.edu

MS31

Exchange Lemma for Nontrivial Slow Flows

The Exchange Lemma of Jones and Kopell deals withthe passage of solutions near a slow manifold of a geo-metric singular perturbation problem x = εf(x, y), y =g(x, y), (x, y) ∈ Rm × Rn. For ε = 0 the slow manifoldis given by g(x, y) = 0 and consists of normally hyperbolicequilibria. For small ε > 0 the slow manifold is nearby, andon it, after a change of variables, the differential equationbecomes x1 = ε, x2 = · · · = xm = 0. However, if nor-mal hyperbolicity fails somewhere, then some fast direc-tions must be added to the slow manifold, and usually theblowing-up technique of Roussarie, Dumortier, Szmolyan,et. al. must be used to analyze the flow on the enlargedslow manifold. I shall discuss generalizations of the Ex-change Lemma needed to study the flow past the enlargedslow manifold. The motivation for this work is rarefaction-like solutions of the Dafermos regularization of a system ofconservation laws.

Stephen SchecterNorth Carolina State UnivDepartment of Mathematicsschecter@math.ncsu.edu

MS32

On the Long Time Regularity of the Shallow WaterEquations

We study the long time existence of classical solutions tothe Shallow Water (SW) equations with pressure gradientand rotational forcing that are both singular within certainscaling regime of the Froude and Rossby numbers. The SW

dynamics is then shown to be asymptotically close to theone governed by the 2-D “pressureless” rotational Eulerequations with subcritical initial data, which in turn yieldsthe increasingly long time existence at this singular regime.The novelty of our approach is the use of an approximatesystem that is linear while still capturing both the singular-ity and the advection dynamics of the underlying nonlinearsystem. The near periodic dynamics shown here is closelyrelated to the circular fluid motions observed in geophysicalsciences.

Bin ChengDepartment of MathematicsUniversity of Maryland at College Parkbincheng@math.umd.edu

Eitan TadmorUniversity of MarylandUSAtadmor@cscamm.umd.edu

MS32

Weakly Nonlinear Approximation of Multi-DHyperbolic-Parabolic System with Entropy

We consider weakly nonlinear approximation around a con-stant state of general multi-D hyperbolic-parabolic systemwith entropy. Using method of multiple time scales, wederive the averaged system. We give a sufficient conditionunder which global weak solution to the averaged systemexists. We apply the results to the multi-D Navier-Stokes-Fourier system of compressible gas dynamics, which sat-isfies the structure condition and derive a new averagedsystem on the fast mode. We apply the global solutionsto the averaged system to construct local Maxwellian toapproximate solutions to the Boltzmann equation

Ning JiangDepartment of MathematicsUniversity of Maryland at College Parknjiang@math.umd.edu

David LevermoreUniversity of Maryland (College Park)USAlvrmr@math.umd.edu

MS32

On the Conditional Global Regularity of the 1DEuler-Poisson Equations with Pressure

We prove that the one-dimensional Euler-Poisson systemdriven by the Poisson forcing together with the usual γ-law pressure admits global solutions for a large class of ini-tial data. Thus, the Poisson forcing regularizes the genericfinite-time breakdown in the 2 × 2 p-system. Global reg-ularity is shown to depend on whether the initial config-uration of the Riemann invariants and density crosses anintrinsic critical threshold.

Dongming WeiUniversity of Maryland, College Parkqiqi@math.umd.edu

Eitan TadmorUniversity of MarylandUSAtadmor@cscamm.umd.edu

280 PD06 Abstracts

MS32

Non-selfsimilar Global Solutions and Their NewStructures of Multi-dimensional ConservationLaws

In this talk, we will discuss the multi-dimensional conser-vation laws whose Riemann data just contain two differentconstant states which are separated by a smooth curve or asurface. Non-selfsimilar structures and properties are dis-covered. Furthermore, global solutions of a class of 2-Dsystems of conservation laws will be also presented and areformulated by implicit function, their structure combining2-D non-selfsimilar elementary waves and non-constant in-termediate states will be shown.

Xiaozhou YangShantou Universityxzyang@stu.edu.cn

MS33

Coherence in SPDE

There has been much recent work in the phenomenon ofstochastic systems which become coherent in certain scal-ing regimes. In this talk, we will present examples of PDEor their corresponding discretizations which act coherentlybut non-trivially under the application of small stochasticforcing.

Lee DevilleCourant Institute of Mathematical Sciencesdeville@cims.nyu.edu

MS33

Noise-induced Target Pattern Formation in Ex-citable Media

Spatially distributed excitable systems, or “excitable me-dia”, are an important class of excitable systems whosemain biological function is long-range signal transmis-sion through self-sustained waves of activity. The stan-dard modeling approach to excitable media is via sys-tems of nonlinear deterministic partial differential equa-tions. These ignore the effect of small random fluctuationsinherent in such media. Here, we show that under rathergeneric conditions the inclusion of the noise may lead toqualitatively new dynamical behaviors, which, neverthe-less, remain essentially deterministic. Specifically, we ana-lyze patterns of recurrent activity in a prototypical modelof an excitable medium in the presence of noise. Withoutnoise, this model robustly predicts the existence of spiralwaves as the only recurrent patterns in two dimensions.Remarkably, we found that, in addition, small noise is ca-pable of generating coherent target patterns, another typeof recurrent activity which is widely observed experimen-tally. Our findings demonstrate the need to re-examinecurrent modeling approaches to active biological media.

Cyrill B. MuratovNew Jersey Institute of TechnologyDepartment of Mathematical Sciencesmuratov@njit.edu

Eric Vanden EijndenCourant InstituteNew York Universityeve2@cims.nyu.edu

Weinan E

Department of MathematicsPrinceton Universityweinan@math.princeton.edu

MS33

Modulation Equations: Stochastic Bifurcation inLarge Domains

We consider the stochastic Swift-Hohenberg equation on alarge domain near its change of stability. We show that,under the appropriate scaling, its solutions can be approxi-mated by a periodic wave, which is modulated by the solu-tions to a stochastic Ginzburg-Landau equation. We thenproceed to show that this approximation also extends tothe invariant measures of these equations.

Greg PavliotisImperial College, Londong.pavliotis@imperial.ac.uk

MS33

Rare Events in Large Systems

Large deviation theory analyzes the probability of a rareevent in a system of bounded size, in the limit of vanish-ing noise. Other limits are also mathematically interestingand physically relevant. In particular, rare events in sys-tems that are large (exponentially large in the inverse noisestrength) raise new questions.

Maria G. ReznikoffPrinceton UniversityMathematics Departmentmrezniko@math.princeton.edu

MS34

Thin Films of Martensitic Materials: Can We Pre-dict Their Microstructure?

Because of their technological applications as microactu-ators, martensitic thin films utilizing the shape memoryeffect have been intensively studied in recent years. Thekey analytical issue for these applications is to provide arigorous derivation of effective thin film models from Non-linear Elasticity via dimension reduction. I will describea new mathematical theory of martensitic thin films anddiscuss its connections with several recent results in thisdirection.

Marian BoceaNorth Dakota State Universitybocea@math.utah.edu

MS34

Atomistic Potential Energy Surfaces and Solid Me-chanics

How does collective “solid mechanics” behaviour emergefrom (accurate quantum-mechanical or approximate)many-atom potential energy surfaces? I will discuss ba-sic open problems, scope and limitation of known methods(e.g. homogenization theory), and recent mathematical re-sults. One such result (joint with F.Theil, Warwick) isidentification of the 3D fcc (face-centered cubic) lattice as alocal minimizer of the celebrated Lennard-Jones pair inter-action model. For a long time any progress in this directionwas beyond reach; a key ingredient is a recent rigidity the-orem from nonlinear PDE theory (due to Friesecke, James,

PD06 Abstracts 281

and Muller).

Gero FrieseckeUniversity of WarwickMathematics Institutegf@maths.warwick.ac.uk

MS34

Mathematical and Computational Methods forCoarse-graining

In this talk we discuss recent progress in obtaining coarse-grained stochastic approximations of microscopic latticedynamics. We prove rigorous error estimates both at equi-librium and non-equilibrium in terms of weak convergenceand relative entropy calculations and obtain a posteriori es-timates that can guide adaptive coarse-grained simulations.We also demonstrate, with direct comparisons between mi-croscopic Monte Carlo and coarse-grained simulations, thatthe derived mesoscopic models provide a substantial CPUreduction in the computational effort.

Markos A. KatsoulakisUMass, AmherstDept of Mathematicsmarkos@math.umass.edu

MS34

Inclusions for Which Constant Magnetization Im-plies Constant Field, and Their Applications

In this paper, we find a class of special regions that have thesame property with respect to second order linear partialdifferential equations that ellipsoids have in infinite space.That is, in physical terms, constant magnetization of theinclusion implies constant magnetic field on the inclusion.The inclusions apparently enjoy many interesting proper-ties with respect to homogenization and energy minimiza-tion. In particular, we use them to give new results on a)homogeneous and inhomogeneous inclusion problems sub-ject to periodic boundary conditions; b) optimal boundsof the effective moduli of two-phase composites; c) energy-minimizing microstructures; and d) the characterization ofthe Gθ-closure of two well-ordered composites.

Liping LiuUniversity of Minnesotaliuliping@aem.umn.edu

Richard JamesUniversity of Minnesota, Minneapolisjames@aem.umn.edu

Perry LeoUniversity of Minnesotaphleo@aem.umn.edu

MS35

Wave Atoms and Time Upscaling of Wave Equa-tions

We present a geometric algorithm for solving the 2D waveequation in smooth inhomogeneous periodic media. Weintroduce a tight frame of ’wave atoms’, a family of wavepackets interpolating between wavelets and Gabor, whosekey property is a precise balance between oscillations andsupport called parabolic scaling. In that frame, the time-dependent Green’s function of the wave equation decom-

poses in a sparse and separable way. As a result, it becomesrealistic to build the full matrix exponential up to sometime which is much bigger than the usual CFL timestep.Once available, this new representation can be used to per-form giant ’upscaled’ time steps to solve the wave equationfaster than with a spectral method. We will show numeri-cal examples as well as complexity results based on a prioriestimates of sparsity and separation ranks.

Laurent DemanetApplied and Computational MathematicsCaltechdemanet@acm.caltech.edu

MS35

Nonlinear Wave Equations on a Curved Back-ground

We will discuss a local well-posedness result for a semilinearwave equation with variable coefficients in 4+1 dimensions.As part of the proof we will present a new definition forthe Xs,b spaces in the variable coefficient case.

Daniel TataruUniversity of California, Berkeleytataru@math.berkeley.edu

Dan A. GebaUniversity of California, Berkeley/ University ofRochesterdangeba@Math.Berkeley.EDU

MS35

Combining Finite Elements and Geometric WavePropagation in 1-D

We show that the initial value problem for a strictly hy-perbolic partial differential equation on the circle can besolved at cost O(N). To obtain this the equation is ap-proximately decoupled, into first order equations that canbe solved using the method of characteristics. An ODE forthe error is of the form dv/dt = R(t)v, with R(t) bounded,which, with an appropriately designed scheme, leads to thestated complexity.

Christiaan C. StolkUniversity of Twentec.c.stolk@ewi.utwente.nl

MS35

Evolution Equations, Curvelet Transform, andCurvelet-curvelet Interaction

We discuss how to solve certain evolution equations withthe curvelet transform. The analysis presented here yieldsan approach that requires solving, scale-wise, for the geom-etry reminiscent of the propagation of singularities on theone hand, and solving a matrix Volterra integral equation(of the second kind) on the other hand. The Volterra equa-tion can be solved by recursion, in a step-by-step mannerwith refinement – as in the computation of certain multiplescattering series; this process reveals the curvelet-curveletinteraction. It is shown that the representation (matrix) ofthe kernel in the Volterra equation is optimally sparse.

Gunther Uhlmann, Hart SmithUniversity of Washingtongunther@math.washington.edu,hart@math.washington.edu

282 PD06 Abstracts

Maarten V. de HoopPurdue Universitymdehoop@purdue.edu

MS36

Numerical Simulation of Vortex Dynamics inGinzburg-Landau-Schrodinger Equation

In this talk, we will study numerically vortex dynamics inthe Ginzburg-Landau-Schrodinger equation (GLSE) in twodimensions (2D)

(α− iβ)∂tψ(x, t) = Δψ +1

ε2

(V0(x) − |ψ|2

)ψ,

x ∈ R2, t > 0,

ψ(x, 0) = ψ0(x), x ∈ R2,

with nonzero far field conditions

|ψ(x, t)| → 1 (e.g. ψ → eimθ), t ≥ 0,

when r = |x| =√x2 + y2 → ∞;

where ψ(x, t) is the complex-valued wave function, V0(x) isa given real-valued external potential satisfying V0(x) → 1when |x| → ∞, ε is a positive constant, α and β are twononnegative constants satisfying α+β > 0. We will presentan efficient and accurate numerical method for numericalsimulation of and apply it to study quantized vortex dy-namics in GLSE. By formulating the equation in 2D polarcoordinate system, the nonzero far field conditions whichare highly oscillating in the transverse direction will be ef-ficiently resolved in phase space in a natural way. Thisallows us to develop a time-splitting method which is un-conditionally stable, efficient and accurate for the prob-lem. Moreover, the method is time reversible when itis applied to the nonlinear Schrodinger equation (NLSE)with nonzero far field conditions. We also apply the nu-merical method to study issues such as the stability ofquantized vortex, interaction of a few vortices, dynamicsof the quantized vortex lattice and motion of vortex withan inhomogeneous external potential in GLSE. Based onour numerical results, we find that the quantized vortexwith winding number |m| = 1 is dynamically stable, andresp., |m| > 1 dynamically unstable, in GLSE. Further-more, different interaction patterns between a few vorticeswith winding number m = ±1 as well as dynamics of quan-tized vortex lattices in GLSE are reported. This talk isbased on joint works with Qiang Du and Yanzhi Zhang.

Weizhu BaoNatl. Univ. of SingaporeDepartment of Mathematicsbao@cz3.nus.edu.sg

MS36

Composite Superconducting Systems

The talk will deal with analytical results regarding surfacenucleation in superconducting samples in contact with asemiconductor or normal material, in the context of theGinzburg-Landau theory. In particular, we will show thatfor a superconductor coated with a properly chosen semi-conductor one should expect enhanced surface supercon-ductivity. In contrast, for an appropriate metal coating,the surface sheet can be completely suppressed.

Tiziana GiorgiDepartment of MathematicsNew Mexico State University

tgiorgi@nmsu.edu

MS36

Capacity of a Multiply-connected Domain andNonexistence of Ginzburg-Landau Minimizers withPrescribed Degrees on the Boundary

Suppose that ω ⊂ Ω ⊂ R2. In the annular domainA = Ω \ ω we consider the class J of complex valuedmaps having the same degree d on ∂Ω and on ∂ω. Theexistence of minimizers of Eκ for all κ when cap(A) ≥ π(domain A is “thin”) and for small κ when cap(A) < π(domain A is “thick”) was established by Berlyand andMironescu (’04). Here we provide the answer for the re-maining case of large κ when cap(A) < π. We prove that,when cap(A) < π, there exists a finite threshold value κ1

of the Ginzburg-Landau parameter κ such that the mini-mum of the Ginzburg-Landau energy Eκ is not attained inJ when κ > κ1 while it is attained when κ < κ1.

Volodymyr RybalkoInstitute for Lower Temperature Physics and Engineerinvrybalko@ilt.kharkov.ua

Dmitry GolovatyThe University of AkronDepartment of Theoretical and Applied Mathematicsdmitry@math.uakron.edu

Leonid BerlyandPenn State UniversityEberly College of Scienceberlyand@math.psu.edu

MS36

On a Minimization Problem with a Mass Con-straint Involving a Potential Vanishing on TwoCurves

We study a singular perturbation type minimization prob-lem with a mass constraint over a domain or a manifold,involving a potential vanishing on two curves in the plane.In the case of the problem on the sphere we give a precisedescription of the limiting behavior of both the minimizersand their energies when the small parameter epsilon goesto zero.

Nelly AndreUniversity of Toursnelly.andre@wanadoo.fr

Itai ShafrirTechionshafrir@math.technion.ac.il

PP0

Spectral-Element Simulations of ElectrokineticFluid Flows in Nanochannels

Micro- and nano-fluidic systems hold tremendous promisefor separation and mixing of biochemical samples. Manysimple devices can be modeled using a two dimensionalapproximation to the governing PDEs for electrokineticfluid flows. Modeling more sophisticated devices, however,will require the simulation of three-dimensional problems,which are computationally challenging to solve using tra-ditional finite-element methods. Spectral elements offer anattactive alternative, especially because many commonly

PD06 Abstracts 283

encountered problem geometries are highly regular. Wepresent spectral-element simulations of several electroki-netic devices.

Jaydeep BardhanElectrical Engineering & Computer ScienceMassachusetts Institute of Technologyjbardhan@mit.edu

Sumita PennathurSandia National Laboratoriessumita@alum.mit.edu

Kevin T. ChuMechanical & Aerospace EngineeringPrinceton Universityktchu@princeton.edu

PP0

An Ant Algorithm for the Multi-Row Layout Prob-lem in Automated/flexiable Manufacturing Sys-tems

Multi-row layout problem is one of commonly used layoutpatterns in which each machine or facility is located in acoordinate system as (x,y). This layout pattern can used inflexible/automated manufacturing systems. In this paper,this problem is formulated as a q.a.p. programming model.An ant algorithm has been developed to solve this problem.An improvement heuristic called local search technique isdeveloped to obtain a betters solution. The performanceof the proposed heuristic is tested over a number of prob-lems selected from the literature. Solving these problemswith proposed algorithm yields better solution comparedto existing algorithm in this area.

Amir JafariUrmia UniversityDepartman of Mechanical EngineeringAMIR JAFARI 2000@YAHOO.COM

PP0

Mathematical Analysis to An Adaptive Network ofthe Plasmodium System

Physarum polycephalum contains a tube network by meansof which nutrients and signals circulate through the body.When the organism is put in a maze and food sources arepresented at two exits, the tube network changes its shapeto connect two exits through the shortest path. Recently,a mathematical model for this adaptation process of path-finding was proposed by Tero et al. We analyze this modelfor some special cases mathematically rigorously.

Isamu OhnishiHiroshima Universityisamu o@math.sci.hiroshima-u.ac.jp

Tomoyuki MiyajiDept. of Math. and Life Sci.,Hiroshima Universitydenpaatama@hiroshima-u.ac.jp

PP0

Viscoelastic Dynamics for a Scalar Two Dimen-sional Continuum Model for Microstructure cre-

ation at an Austenite-Martensite Interface

A pseudo-spectral Fourier-Chebyshev collocation methodfor simulating a scalar two-dimensional viscoelastic dy-namic model of microstructure creation with capillarity,utt − βΔut = (3u2

x − 1)uxx + uyy − εΔ2u, is developedand implemented in Matlab. By using an integration ma-trix formulation for the Chebyshev modes, the numericalinstability associated with large dimension Chebyshev dif-ferentiation matrices is overcome. The integration matricesare sparse and so allow a large number of modes to be com-puted, and hence for small values of the spatially regular-izing capillarity parameter to be used. The computationalresults are compared to previous simulations without cap-illarity.

Benson K. MuiteUniversity of Oxfordmuite@maths.ox.ac.uk

PP0

Symbolic Integration and Summation

The homotopy algorithm is a powerful method for indefi-nite integration of total derivatives, and for the indefinitesummation of differences. By combining these ideas withstraightforward Gaussian elimination, we construct an al-gorithm for the optimal symbolic integration or summationof expressions that contain terms that are not total deriva-tives or differences. The optimization consists of minimiz-ing the number of terms that remain unintegrated or arenot summed. Further, the algorithm imposes an orderingof terms so that the differential or difference order of theseremaining terms is minimal.

Bernard Deconinck, Michael A. NivalaUniversity of Washingtonbernard@amath.washington.edu,man9@amath.washington.edu

PP0

The Billiard Problem in Nonlinear and Nonequi-librium System

We consider about the billiard problem in nonlinear andnonequilibrium system, whose behavior is similar to theusual billiard problem at a glance. However it has someremarkable properties quite different from the usual one inview of dynamical system. We reveal some of them by useof a reduction equation. We finally show that the strangebehavior comes from intermittent type Chaos possessed bythe system by use of computer simulation.

Isamu OhnishiHiroshima Universityisamu o@math.sci.hiroshima-u.ac.jp

Tomoyuki MiyajiDept. of Math. and Life Sci.,Hiroshima Universitydenpaatama@hiroshima-u.ac.jp

PP0

Oil Production Linked to Stochastic DifferentialEquation

Oil barrel price is a random variable that can be modeledby a stochastic differential equation. Here we illustrate asimple model for the net cash of the oil production based on

284 PD06 Abstracts

the oil well production, oil barrel price, production costs,initial costs, and the decline production in order to calcu-late the production to maximize the net cash. The opti-mization problem is solved by two approaches: a discretemethod and one based on Markov controls.

David E. Rivera-RecillasInstituto Mexicano del Petroleodrivera@imp.mx

Francisco Rocha-LegorretaInstituto Mexicano del Petroleofrocha@imp.mx

Israel Castillo-NunezInstituto Mexicano del Petroleoiacastil@imp.mx

PP0

Double Reduction from the Association of Symme-tries with Conservation Laws: Applications to theHeat, Bbm, and One Dimensional Gas DynamicsEquations

A method to find a double reduction of partial differen-tial equations (PDEs) with two independent variables isderived. The method is based on the definition of the as-sociation of Lie point symmetries of PDEs with conserva-tion laws [Kara A.H. and Mahomed F.M.,The relationshipbetween symmetries and conservation laws, Int. J. Theo-retical Phys. 39(1)(2000) pp24-30].

Applications to the linear heat equation,the BBM equationand a system of differential equations from one dimensionalgas dynamics illustrate the new theory.

Astri SjobergUniversity of Johannesburg,Department of Applied Mathematicsasj@na.rau.ac.za

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A Quasistationary Analysis of a Stochastic Chem-ical Reaction: Keizer’s Paradox Revisited

The deterministic and stochastic models of an autocat-alytic biochemical reaction are compared. The two modelsshow distinctly different steady state behavior. To resolvethis “paradox” proposed by J. Keizer, we compute the ex-pected time to extinction and a quasistationary probabilitydistribution in the stochastic model. Mathematically, weidentify that exchanging the limits of infinite system sizeand infinite time is problematic. An appropriate systemsize for numerical computations is discussed.

Hong QianDepartment of Applied MathematicsUniversity of Washingtonqian@amath.washington.edu

Melissa M. VellelaUniversity of WashingtonApplied Mathematicstmbgnut@gmail.com