2017 Workshop on Nonlinear PDEs inApplied Mathematics

Izmir Institute of Technology, Izmir, TURKEY

August 8-10, 2017

ii

Support

We thank the following organizations for their support:

Members of the organizing and scientific committee: Varga Kalantarov(Koc), Turker Ozsarı (IZTECH), Edriss Titi (Texas A&M), Irena Lasiecka (Mem-phis), Andrei V. Fursikov (Moskow State), Mete Soner (ETH), Sergey Zelik (Surrey),Roberto Triggiani (Memphis).

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iv

Contents

Support iii

Schedule 1

Abstracts 3

Boundary effect in the zero viscosity limit and energy dissipation (ClaudeBardos*, Laszlo Szekelyhidi, E. Titi) . . . . . . . . . . . . . . . . . . 4

The Resonant Soliton Hierarchies. From RNLS and KP-II to Relativisticand q-Dispersive Resonant Solitons (Oktay Pashaev* ) . . . . . . . . . 5

On nonlinear damped wave equations for positive operators (Niyaz Tok-magambetov* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Vector-valued Phase-field Models of Obstacle Type (Orestiz Vantzos* ) . . 7

TBA (Nader Masmoudi* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Schrodinger equation in QFD representaiton with Navier - Stokes Dissipa-tion (Attila Askar* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Remark on the eigenvalues of the m-Laplacian in a punctured domain(Gulzat Nalzhupbayeva* ) . . . . . . . . . . . . . . . . . . . . . . . . . 12

Continuous Data Assimilation for the Two Dimensional g-Benard Problem(Meryem Kaya, Muharrem Ozluk* ) . . . . . . . . . . . . . . . . . . . 13

Global stabilization of solutions to the complex Ginzburg-Landau equationby finite parameter controllers (Jamila Kalantarova*, Turker Ozsarı) 14

A new modified schemes for shallow water linear equations and Burgersnonlinear equation (Abdelhamid Laouar* ) . . . . . . . . . . . . . . . 15

Existence of Minimal and Maximal solutions for Quasilinear Elliptic Equa-tion with Nonlocal Boundary Conditions on Time-Scales (MohammedDerhab, Mohammed Nehari* ) . . . . . . . . . . . . . . . . . . . . . . 16

Description of a New Generalized Quasi Two-Phase Mass Flow Model(Puskar R. Pokhrel* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Global Well-posedness of an Inviscid Three-dimensional Pseudo-Hasegawa-Mima-Charney-Obukhov Model (Chongsheng Cao, Aseel Farhat, EdrissS. Titi* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Comparison of solutions of some families of wave equations (Husnu AtaErbay, Saadet Erbay, Albert Erkip* ) . . . . . . . . . . . . . . . . . . 19

Baroclinic Instability for 3D Boussinesq Flows (Mustafa Taylan Sengul*,Shouhong Wang) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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Inertial manifolds for reaction-diffusion-advection problems (Anna Kos-tianko* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Hyperbolic Relaxation of the 2D Navier-Stokes Equations in a BoundedDomain (Sergey Zelik* ) . . . . . . . . . . . . . . . . . . . . . . . . . 22

Local and Global Lyapunov Exponents Revisited (Alp Eden* ) . . . . . . . 23Determination of the Wall Function for Navier-Stokes Solutions on Carte-

sian Grids (Emre Kara* ) . . . . . . . . . . . . . . . . . . . . . . . . . 24An Active Control of a Double-Beam Complex System (Melih Cınar* ) . . 25Well-posedness of the fourth-order Schrodinger equation on the half-line in

fractional Sobolev spaces (Turker Ozsarı, Nermin Yolcu* ) . . . . . . 26Analytical solution of the Volterra-Fredholm singular integro-differential

(Fernane Khaireddine* ) . . . . . . . . . . . . . . . . . . . . . . . . . 27Non-Homogeneus Continuous and Discrete Gradient Systems: The Quasi-

convex case (Vahid Mohebbi* ) . . . . . . . . . . . . . . . . . . . . . . 29Existence and Continuation Theorems of Caputo Type Fractional Differ-

ential Equations (Shahzad Sarwar* ) . . . . . . . . . . . . . . . . . . . 30Global uniqueness and stability of an inverse problem for the Schrodinger

equation on a Riemannian manifold via one boundary measurement(Roberto Triggiani* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Error analysis of a semi-discrete method for nonlocal nonlinear wave equa-tions (Husnu Ata Erbay*, Saadet Erbay, Albert Erkip) . . . . . . . . . 32

Stability estimates for some inverse problems for ultrahyperbolic Schrodingerequations (Fikret Golgeleyen*, Ozlem Kaytmaz ) . . . . . . . . . . . . 33

Long-time behaviour of the strongly damped semilinear plate equation inRn (Sema Yayla* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Small Data Global Existence and Exponential Decay of Solutions to a FreeBoundary Fluid-Structure Interaction System with Damping (Mi-haela Ignatova, Igor Kukavica, Irena Lasiecka, Amjad Tuffaha* ) . . . 36

Complex Ginzburg-Landau equations with dynamic boundary conditions(Wellington Jose Correa, Turker Ozsarı* ) . . . . . . . . . . . . . . . 38

The Stability of an IMEX (Implicit-Explicit) Method for Simplified MHD(MagnetoHydroDynamics) Equations (Simge Kacar Eroglu*, GamzeYuksel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Inverse Source Problem for Benjamin-Bona-Mahony-Burger Equation (SerapGumus* ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Boosting the decay of solutions of the Korteweg-de Vries-Burgers equationto a predetermined rate from the boundary (Eda Arabacı*, TurkerOzsarı) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

TBA (Varga Kalantarov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Author Index 43

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Schedule

1

2

Abstracts

* refers to the person who will be delivering the talk.

3

Boundary effect in the zero viscosity limit and energydissipationAugust 8th

9:10-10:00Claude Bardos*, Laszlo Szekelyhidi, E. Titi

University of Paris 7- Denis Didero

This talk is based on the two following observations. First the notion of weekconvergence is the deterministic avatar of the notion of average in the statisticaltheory of turbulence. Second in most cases (even for generation of homogeneous andisotropic turbulence) , the basic effects are due to the boundary. It turns out that itis a theorem of T. Kato of 1984 that allows through diverse generalizations (objectof this presentation) to make a natural connection between these two observations;moreover what is striking is that the hypothesis of this theorem are in full agreementwith observations coming from engineering science or fluid mechanic. Eventually thefact that (even in domain with boundary) weak solutions of the Euler equations areof constant energy as soon as they belong to an Holder space C0,α(Ω) with α > 1

3

(the so called Onsager’s conjecture) leads to a comparison between the Kolmogorov1/3 law and the Kato hypothesis for the absence of energy dissipation.

4

The Resonant Soliton Hierarchies. From RNLS and KP-IIto Relativistic and q-Dispersive Resonant Solitons August 8th

10:00-10:50Oktay Pashaev*

Izmir Institute of Technology

In this talk I review integrable resonant soliton hierarchies and describe some newresults and applications. I will start from SL(2, R) AKNS hierarchy, the second flowof which is related with resonant NLS (RNLS) equation in 1 + 1 dimensions. Thisequation appears in cold plasma physics and includes de Broglie-Bohm quantumpotential. Soliton interactions for RNLS equation show the resonance phenomenaand in non-Madelung fluid representation it is related with the Broer-Kaup reso-nant system. The Malomed-Stenflo generalized NLS as shown, in special domain ofparameters admits soliton resonances and can be transformed to RNLS. Combiningthe second and the third flows of the hierarchy we construct resonant line solitons ofKP − II equation in 2 + 1 dimensions. As a next natural generalization we discussthe resonant DS equation, 2 + 1 dimensional nonlocal RNLS and corresponding2+1 dimensional BroerKaup type system. By the recursion operator of AKNS hier-archy, the relativistic and q-dispersive resonant soliton equations are derived on thishierarchy. Similar approach is applied to the Kaup-Newell hierarchy and resonantDNLS equation with chiral soliton interactions. The second and the third flowsfor this hierarchy describe resonant solitons of MKP-II. By the recursion operatorthe relativistic resonant DNLS equation is constructed and q-dispersive resonantequations are discussed.

This work is supported by TUBITAK Grant 116F206.

Bibliography

[1] Pashaev O.K., Lee J.-H., Resonance solitons as black holes in Madelung fluid,Mod. Phys. Lett. A17 (2002), 1601-16-19

[2] Lee J.-H. et al. The resonant nonlinear Schrodinger equation in cold plasmaphysics, J. Plasma Physics 73 (2007) 257-272

[3] Pashaev O.K., Relativisitc Burgers and nonlinear Schrodinger equations, Theor.Math. Phys. 160 (2009) 1022-1030

[4] Rogers C., Pashaev O.K., On a 2+1 dimensional Whitham-Broer-Kaup system:a resonant NLS connection, Stud. Appl. Math. 127 (2011) 141-152

[5] Pashaev O.K., Lee J.-H., Relativisitc DNLS and Kaup-Newell hierarchy, SIGMA13 (2017)058

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On nonlinear damped wave equations for positive operatorsAugust 8th11:10-11:35

Niyaz Tokmagambetov*

Al-Farabi Kazakh National University

In this research we study a Cauchy problem for the nonlinear damped waveequations for a general positive operator with discrete spectrum. We derive the ex-ponential in time decay of solutions to the linear problem with decay rate dependingon the interplay between the bottom of the operator’s spectrum and the mass term.Consequently, we prove global in time well-posedness results for semilinear and formore general nonlinear equations with small data. Examples are given for nonlin-ear damped wave equations for the harmonic oscillator, for the twisted Laplacian(Landau Hamiltonian), and for the Laplacians on compact manifolds.

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Vector-valued Phase-field Models of Obstacle Type August 8th11:35-12:00

Orestiz Vantzos*

Technion - Israel Institute of Technology

We present a class of phase-field models based on the double-obstacle Ginzburg-Landau functional. They involve vector-valued potentials with values constrainedon convex sets beyond the classic Gibbs complex, and are therefore suitable forthe modeling of multi-phase problems with various complicated domain/junctionarrangements. We discuss their efficient variational discretization, based on DeGiorgi’s minimizing movements, and certain interesting geometrical insights con-cerning their behavior in the sharp limit.

Bibliography

[1] Blowey, J. F. and Elliott, C. M., The Cahn–Hilliard gradient theory for phaseseparation with non-smooth free energy Part II: Numerical analysis., EuropeanJournal of Applied Mathematics 2.03, 233-280 (1991).

[2] Blowey, J. F. and Elliott, C. M., The Cahn–Hilliard gradient theory for phaseseparation with non-smooth free energy Part I: Mathematical analysis., EuropeanJournal of Applied Mathematics 3.02, 147-1790 (1992).

[3] De Giorgi, Ennio., New problems on minimizing movements., Ennio de Giorgi:Selected Papers, 699-713, (1993).

[4] Garcke, H., Nestler, B., and Stoth, B., A multiphase field concept: numericalsimulations of moving phase boundaries and multiple junctions., SIAM Journalon Applied Mathematics 60.1, 295-315 (1999).

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TBAAugust 8th13:00-13:50

Nader Masmoudi*

New York University

To be announced...

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Schrodinger equation in QFD representaiton with Navier -Stokes Dissipation August 8th

13:50-14:40Attila Askar*

Koc University

1. Introduction. The Quantum Fluid Dynamics (QFD) representation ofquantum mechanics was partly motivated by questions of the completeness of thequantum theory, while admitting to its internal consistency. The QFD representa-tion has its foundations in Madelung’s work (1926) in the early times and followedby Bohm’s interpretation of quantum mechanics (1950’s) with the goal to find clas-sically identifiable dynamical variables at the sub-particle level [1]. The approachleads to conservation laws for ”mass”, ”momentum” and ”energy” similar to thosein hydrodynamics for a compressible, non dissipative fluid.

The QFD equations are a set of nonlinear partial differential equations. Inthis sense, they may be seen as a step in the negative direction as compared tothe linear Schrodinger equation. However, in this scheme, the oscillatory real andimaginary components of the complex wave function are replaced by the monotonousthe amplitude and phase. This has been exploited as a significant advantage incomputational quantum mechanics [1, 2].

In this study, the nonlinear field equations are suggested as the natural frameworkfor studying fundamentally nonlinear phenomena such as solitons and chaos, in amulti-dimensional context. This expose extends the QFD formalism to the case ofa non-isolated medium. The interaction of this analogous fluid medium with theenvironment is postulated to lead to a dissipative piece in the acceleration and isintroduced phenomenologically as in the classical Navier - Stokes equation.

The Schrodinger equation being linear and having basic conservation laws formomentum and energy does not show possibilities for ”turbulence”. As for theNavier - Stokes equation, this formulation ”may” contain a necessary ingredient ofchaotic behavior for quantum mechanics. This modified form allows for vorticitiesthat may break into a cascade, eventually leading to ”turbulence”, as in the classicalNavier - Stokes equation.

2. From Schrodinger’s to Quantum Fluid Dynamics (QFD) Equation.The starting point is the Schrodinger equation for the complex valued wave func-tion ψ, nondimensionalized by reducing the Planck’s constant ~ of the conventionalquantum mechanics and the mass m of the particle to unity is:

i~∂ψ

∂t= −1

2∇2ψ + V ψ

V = V (x, t) is the externally applied potential energy that guides the dynamicsand i =

√−1 is the usual imaginary unit.

The complex wave function ψ is represented by its amplitude A and phase S as:

ψ(x, t) = A(x, t) ei[S(x,t)−Et]

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Above by reference to quantum mechanics, E is singled out to be identified asat the total energy in the derivations below.

The substitution of the representation of the wave function in the polar form inthe Schrodinger equation, the separation of its respectively imaginary and real partsand the definition of the density ρ, velocity v, the externally applied force g = −∇Vand the ”Quantum potential” or ”pressure” yield the hydrodynamics representationof quantum mechanics. The equations are respectively ”Mass conservation” and”Momentum conservation” equations:

ρ = A2 v =∇Sm

Vq = −1

2

∇2A

Ag = −∇V

∂ρ

∂t+∇ · (ρv) = 0

∂v

∂t+ v.∇v = −∇Vq + g

These are called alternatively as ”Madelung’s quantum hydrodynamics equa-tions”, ”Bohmian representation of quantum mechanics”, ”Quantum fluid dynamics(QFD) equations”...

The energy equation is also derived as an alternative to the momentum equation:

∂S

∂t+

1

2v · v + V + Vq = E

S is interpreted as an action whose time rate St is an energy; 12mv · v + V form

the classical energy with its kinetic and potential energy components; and Vq as anon-local ”quantum potential” as the Laplacian connects neighboring regions.

The linear and nonlinear parts of the momentum equation are the local andconvective acceleration terms of classical fluid continua as in the Navier - Stokesequations. The acceleration in the Navier - Stokes equation contains also a duffi-sive component represented by the phenomenological term −µ∇2v. µ is callesd as”viscosity”. Adding this term to the momentum equation above leads to:

∂v

∂t+ v.∇v = −∇(V + Vq)− µ∇2v

3. Further use of QFD. The same decomposition of the wave funtion canalso be applied to the Nonlinear Schrodinger equation that leads to the same massconservation equation and slightly modified momentum and energy equations:

i∂ψ

∂t= −1

2∇2ψ − k2|ψ|2ψ) → ∂v

∂t+ v.∇v = −∇(Vq + k2ρ)

∂S

∂t+ (

1

2|v|2 + Vp + k2ρ) = E

QFD allows also the representation Gross - Pitaevskii equation for Bose-Einsteincondensation.

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References

[1] (Excellent review of the field) Robert E. Wyatt, Quantum Dynamics withTrajectories: Introduction to Quantum Hydrodynamics, Springer-Verlag NY, 2005.

[2] (Sample of the Work by AA’s contributions over the years)J. H. Weiner and A. Askar, J. Chem. Phys. 54, 1108 (1971).

A. Askar and J. H. Weiner, Am. J. Phys. 39, 1230 (1971).

A. Askar and M. Demiralp, J. Chem. Phys. 60, 2762 (1974)

A. Askar, A. S. Cakmak and H. Rabitz, J. Chem. Phys. 72, 5287 (1980).

B. Dey, A. Askar and H. A. Rabitz, J. Chem. Phys. 109, 8770 (1998)

F. Sales, A. Askar and H. A. Rabitz, J. Chem. Phys. 11, 2423 (1999)

11

Remark on the eigenvalues of the m-Laplacian in apunctured domainAugust 8th

15:00-15:25Gulzat Nalzhupbayeva*

Al-Farabi Kazakh National University

In this research we extend results on regularized trace formulae which were es-tablished by Kanguzhin and his co-authors for the Laplace and m-Laplace operatorsin a punctured domain with the fixed iterating order m ∈ N. By using techniquesof Sadovnichii and Lyubishkin, Kanguzhin and his co-authors described regular-ized trace formulae in the spatial dimension d = 2. In this remark one is to beclaimed that the formulae are also valid in the higher spatial dimensions, namely,2 ≤ d ≤ 2m. Also, we give the further discussions on a development of the analysisassociated with the operators in punctured domains. This can be done by using socalled ’nonharmonic’ analysis.

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Continuous Data Assimilation for the Two Dimensionalg-Benard Problem August 8th

15:25-15:50Meryem Kaya, Muharrem Ozluk*

Gazi university

Data assimilation refers to the process of completing, or enhancing the resolu-tion of the initial condition. The idea behind continuous data assimilation is toassimilate observational data into a model as it is being integrated in time so thatan approximate solution converges to the true solution [1, 2]. In this talk we intro-duce an algorithm for continuous data assimilation for the two-dimensional g-Benardconvection problem by following Farhat, Jolly and Titi [3].

Bibliography

[1] E. Olson, E. S. Titi (2003). Determining modes for continuous data assimilationin 2D turbulence. J. Stat. Phys. 113(516), 799-840.

[2] E. Olson, E. S. Titi (2008). Determining modes and Grashof number in 2Dturbulence. Theor. Comput. Fluid Dyn., 22(5), 327-339.

[3] A. Farhat, M. S. Jolly and E. S. Titi (2015). Continuous data assimilation forthe 2D Benard convection through velocity measurements alone. Physica D. 303:59-66.

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Global stabilization of solutions to the complexGinzburg-Landau equation by finite parameter controllersAugust 8th

15:50-16:15Jamila Kalantarova*, Turker Ozsarı

Izmir Institute of Technology

This talk concerns the initial boundary value problem for the complex Ginzburg-Landau equation:

ut − (λ+ iα)∆u+ (κ+ iβ)|u|pu− γu = Π(u), x ∈ Ω, t > 0,

where u denotes the complex oscillation amplitude, β ∈ R and α ∈ R are the(nonlinear) frequency and (linear) dispersion parameters, respectively. The con-stants λ and κ are assumed to be strictly positive. Ω ⊂ Rn is a bounded domainand p > 0 is the source-power index. The operator Π at the right hand side denotesa finite parameter feedback controler, which employ finitely many volume elements,Fourier modes or nodal observables. We prove the global exponential stabilizationof this problem. Moreover, steering solutions of the Ginzburg-Landau equation to adesired solution of the non-controlled system employing finite parameter controllersis established.

Both researchers in this work were supported by TUBITAK 3501 Career Grant115F055

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A new modified schemes for shallow water linear equationsand Burgers nonlinear equation August 8th

16:14-16:40Abdelhamid Laouar*

Badji Mokhtar University of Annaba

This work concerns the study of the linear equations of the shallow water andthe nonlinear equation of Burgers for a perfect fluid and a weakly viscous fluid. Forthe equations of shallow water, the phenomenon is modeled by a system of improvedequations of Boussinesq. We add to them the numerical dispersion due to truncationerrors (discretization errors). For the solution, we adopt the finite difference methodcombined with the implicit scheme of the alternate direction. In the same way, weproceed for the nonlinear equation of Bourgers. Then, we study the stability andthe convergence of the solution and make a comparison with some existing results.Finally for illustration, we present numerical simulations.

Keywords: Boussinesq equations, Bourgers equation, Modified schemes, Sta-bility and convergence.

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Existence of Minimal and Maximal solutions for QuasilinearElliptic Equation with Nonlocal Boundary Conditions on

Time-ScalesAugust 8th16:40-17:05

Mohammed Derhab, Mohammed Nehari*

E.S.M.Tlemcen

Abstract. The purpose of this work is the construction of minimal and maximalsolutions for a class of second order quasilinear elliptic equation subject to nonlocalboundary conditions. More specifically, we consider the following nonlinear bound-ary value problem −

(ϕp(u∆))∆

= f (x, u) , (a, b)T ,u (a)− a0u

∆ (a) = g0 (u) ,u (σ (b)) + a1u

∆ (σ (b)) = g1 (u) ,

(1)

where p > 1, ϕp (y) = |y|p−2 y,(ϕp(u∆))∆

is the one-dimensional p−Laplacian,f : [a, b]T × R → R is a rd-continuous function, gi : Crd ([a, b]T )× Crd ([a, b]T ) → R(i = 0 and 1) are rd-continuous and a0 and a1 are a positive real numbers.

Key words: Quasilinear elliptic equation; Time-Scale, Nonlocal boundary con-ditions; upper and lower solutions; monotone iterative technique

AMS Classification: 34B10, 34B15

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Description of a New Generalized Quasi Two-Phase MassFlow Model August 8th

17:05-17:30Puskar R. Pokhrel*

Kathmandu University

Employing the full dimensional two-phase (mixture of solid particles and viscousfluid) mass flow model equations [1], we have generated a novel, generalized quasitwo-phase, full two-dimensional model for bulk mixture flow down a channel. Theemerging model is written as a well-structured system of three highly non-linearpartial differential equations in conservative form representing the mass and themomentum balances. The new mechanical and dynamical concepts of generalizedbulk and shear viscosities, pressures, and velocities for the mixture characterize themodel. Some new reduced models are also obtained by considering different aspectsof dynamics and modeling. The advantage of the model lies mainly in providinga possibility for simulating the mixture velocities and pressures much faster thanthe two-phase mass flow model. Furthermore, the introduction of the velocity andpressure drifts factors makes it possible to reconstruct the two-phase mass flow soas to capture its basic dynamics.

Keywords: Quasi-two-phase flow, generalized bulk and shear viscosities, Gen-eralized mixture velocities and pressure, Velocity and pressure drift factors

Bibliography

[1] Pudasaini SP (2012) A general two-phase debris flow model. J. Geophys. Res.117, F03010, doi: 10.1029/ 2011JF002186.

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Global Well-posedness of an Inviscid Three-dimensionalPseudo-Hasegawa-Mima-Charney-Obukhov ModelAugust 9th

9:10-10:00Chongsheng Cao, Aseel Farhat, Edriss S. Titi*

The Weizmann Institute of Science and Texas A&M University

The 3D inviscid Hasegawa-Mima model is one of the fundamental models thatdescribe plasma turbulence. The same model is known as the Charney-Obukhovmodel for stratified ocean dynamics, and also appears in literature as a simpli-fied reduced Rayleigh-Benard convection model. The mathematical analysis of theHasegawa-Mima and of the Charney-Obukhov equations is challenging due to thetheir resemblance with the Euler equations. In this talk, we introduce and show theglobal regularity of a model which is inspired by the inviscid Hasegawa-Mima andCharney-Obukhov models, named a pseudo-Hasegawa-Mima model. The introducedmodel is easier to investigate analytically than the original inviscid Hasegawa-Mimamodel, as it has a nicer mathematical structure. To establish our global regularityresult we implement a new logarithmic inequality, generalizing the Brezis-Gallouet-Berzis-Wainger inequalities.

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Comparison of solutions of some families of wave equations August 9th10:00-10:50

Husnu Ata Erbay, Saadet Erbay, Albert Erkip*

Sabancı University

We will present a series of recent results concerning the comparison of solutionsof two different wave-type equations in asymptotic regimes determined by two pa-rameters characterising long waves and small amplitude. There are many works inliterature on similar comparisons within the scope of fluid dynamics. These com-parisons are made between a model equation, typically the Camassa-Holm (CH)equation, and a parent equation, namely the Euler equation, the Boussinesq sys-tem, or similar. In our work, we have considered the same question within the scopeof nonlocal elasticity. In that respect we first show that the CH equation can beformally derived from the improved Boussinesq (IB) equation. We then give a rig-orous justification of this, namely prove that solutions of the CH equation are wellapproximated by appropriate solutions of the IB equation. Conversely we show thatsolutions of the IB equation can be written as a sum of right and left going solutionsof the CH equations up to a small error. These results can be generalised to nonlocalwave equations which have a similar dispersion relation as the IB equation in thelong wave limit. Finally we consider the comparison of nonlocal equations and giveestimates between solutions of two nonlocal wave equations.

Bibliography

[1] H. A. Erbay, S. Erbay and A. Erkip, The Camassa-Holm equation as the long-wave limit of the improved Bousssinesq equation and of a class of nonlocalwave equations, Discrete and Continuous Dynamical Systems - Series A, 36(2016),6101–6116.

[2] H. A. Erbay, S. Erbay and A. Erkip, On the decoupling of the improved Boussi-nesq equation into two uncoupled Camassa-Holm equations, Discrete and Con-tinuous Dynamical Systems - Series A, 37(2017), 3111-3122

19

Baroclinic Instability for 3D Boussinesq FlowsAugust 9th11:10-11:35

Mustafa Taylan Sengul*, Shouhong Wang

Marmara University

The main aim of this talk is to describe the stability and dynamic transitions ofa basic shear flow, associated with geophysical baroclinic instability, for the threedimensional (3D) continuously stratified rotating Boussinesq model. The modeladmits a steady state solution characterizing a shearing motion which, due to theCoriolis forces, must be balanced by a pressure gradient in response to spatiallyvarying density field. We first establish thresholds for the energy stability and linearstability of this basic solution via energy method. Next by numerically computingthe center manifold using a spectral method, we establish the reduced equations ofthe system at the onset of transition and prove the existence of various transitiontypes describing transitions to multiple steady states as well as to spatio temporaloscillations. This is joint work with Shouhong Wang.

20

Inertial manifolds for reaction-diffusion-advection problems August 9th11:35-12:00

Anna Kostianko*

University of Surrey

An inertial manifold (IM) is one of the key objects in the modern theory ofdissipative systems generated by partial differential equations (PDEs) since it al-lows us to describe limit dynamics of the considered system by the reduced finite-dimensional system of ordinary differential equations (ODEs). It is well known thatthe existence of an IM is guaranteed when the so called spectral gap conditions aresatisfied, whereas their violation leads to the possibility of an infinite dimensionallimit dynamics, at least on the level of an abstract parabolic equation. However,these conditions restrict greatly the class of possible applications and are usuallysatisfied in the case of one spatial dimension only.

The main aim of this talk is a comprehensive study of reaction-diffusion-advection(RDA) equations in 1D with different boundary conditions (BC). Even in 1D casethe spectral gap condition does not satisfied for RDA equations and therefore ques-tion on existence/non-existence of an IM for it requires accurate analysis. In thecase of Dirichlet or Neumann boundary conditions we will show the existence of anIM using a specially designed non-local in space diffeomorphism which transformsthe equations to the new ones for which the spectral gap conditions are satisfied.In contrast to this, in the case of periodic boundary conditions, we will construct anatural example of a RDA system which does not possess an IM. It is worth men-tioning that scalar RDA equation even with periodic BC possesses IM, and this maybe shown by using the modification of the transform proposed for Dirichlet BC.

21

Hyperbolic Relaxation of the 2D Navier-Stokes Equationsin a Bounded DomainAugust 9th

13:00-13:50Sergey Zelik*

University of Surrey

A hyperbolic relaxation of the classical Navier-Stokes problem in 2D boundeddomain with Dirichlet boundary conditions is considered. It is proved that thisrelaxed problem possesses a global strong solution if the relaxation parameter issmall and the appropriate norm of the initial data is not very large. Moreover, thedissipativity of such solutions is established and the singular limit as the relaxationparameter tends to zero is studied.

22

Local and Global Lyapunov Exponents Revisited August 9th13:50-14:40

Alp Eden*

Emeritus from Bogazici University

A topological theory of local and global Lyapunov exponents for abstract dy-namical systems was introduced by Eden, Foias and Temam [2] in the late 1980’s.Although the theory was developed in order to obtain better dimension upper esti-mates for the global attractors for dissipative and/or damped PDE’s, it had moreconclusive results for low dimensional dynamical systems such as the Lorenz systemof ODE’s. Three questions were posed in a conference in Luminy in 1989 seems tohave led to some interesting research.[1] A series of results by Leonov in 1990’s ledto the positive resolution of the third question about the local Lyapunov dimensionof the Lorenz attractor. [3]

There has been a renewed interest in the field by Leonov, Kuznetsov and co. (see e.g. [4, 5]) In my talk, I will take a nostalgic journey from the beginning of thistheory to some of the more recent results. The talk will not be technical.

Bibliography

[1] Eden, A, “Local Lyapunov exponents and a local estimate of Hausdorff dimen-sion”, Attractors, inertial manifolds and their approximation (Marseille-Luminy,1987). RAIRO Model. Math. Anal. Numer.23 (1989), no. 3, 405–413.

[2] Eden, A, ; Foias, C. and Temam, R., “Local and Global Lyapunov Exponents ”, Journal of Dynamics and Differential Equations, 3, 133-177, 1991.

[3] Leonov, G.A. and Lyashko, S. A., “On Eden’s conjecture for Lorenz System”,1993, Vestnik, vol. 1, no.3, 9-15.

[4] A Yu Pogromsky, G Santoboni and H Nijmeijer , “An ultimate bound on the tra-jectories of the Lorenz system and its applications”, 20 June 2003 • Nonlinearity,Volume 16, Number 5.

[5] Leonov, G. A. and Kuznetsov, N. V., “A short survey on Lyapunov di-mension for finite dimensional dynamical systems in Euclidean space”, 2016,arXiv:1510.03835.

23

Determination of the Wall Function for Navier-StokesSolutions on Cartesian GridsAugust 9th

15:00-15:20Emre Kara*

University of Gaziantep

Cartesian grids have beneficiary effects in the solution of partial differential equa-tions (PDEs) resulting from their characteristic form of discretized finite-volumerepresentation, i.e. nonlinear Navier Stokes equations involving irregular and mul-tielement boundaries.

A new flow solver is generated in Visual Studio by using object-oriented FOR-TRAN programming and is called GeULER-NaTURe (cartesian Grid generator withEULER Navier Stokes TURbulent flow solvEr) [1]. Viscous terms of Navier-Stokesequations and Spalart-Allmaras [2] turbulence model are implemented into solver.

Near the solid boundary, generally, wall function is established by universal law-of-the-wall coordinate, y+ , presented by Spalding [3].

GeULER-NaTURe flow solver [1] is dealing with external turbulent flows withhigh Reynolds numbers around embedded boundaries, so that the wall-bounded flowapproach should be followed to limit the resolution of the fine grids in the turbulentboundary layer. In this region, the wall function is employed for the coarse gridutilization at 30 ≤ y+ ≤ 150 instead of fine grids at y+ < 5 thus eliminates stiffnessproblem owing to redundant refinement. The analytic solution suggested by Bergeret al. [4] in place of Spalding formula is selected as the wall function predictor,since the proposed formula is an explicit function of velocity in contrast to Spaldingformula. Turbulent viscosity required for SA relations is adapted from Frink’s study[5].

Bibliography

[1] Kara, E. Development of a Navier Stokes solver for compressible flows on Carte-sian grids with aerodynamics applications. Doctoral Dissertation, University ofGaziantep, Gaziantep, 2015.

[2] Spalart, P. R., Allmaras, S. R. A one-equation turbulence model for aerodynamicflows. AIAA Paper, 92–0439, 1–16, 1992.

[3] Spalding, D. B. A single formula for the “law of the wall”. Journal of AppliedMechanics, 28(3), 455–458, 1961.

[4] Berger, M. J., Aftosmis, M. J., Allmaras, S. R. Progress towards a Cartesiancut-cell method for viscous compressible flow. AIAA Paper 2012, 1301, 1–24,2012.

[5] Frink, N. T. Assessment of an unstructured-grid method for predicting 3-D tur-bulent viscous flows. In: 34th AIAA Aerospace Sciences Meeting, AIAA Paper,292, 1–12, 1996.

th Shouhong Wang.

24

An Active Control of a Double-Beam Complex System August 9th15:25-15:50

Melih Cınar*

Yildiz Technical University

This paper deals with an active optimal control of free transverse vibrations of anelastically bonded continuous beam-string system. The aim of the paper is to min-imize the physical energy through displacement and the velocity of the system. Forthis purpose, the actuators are placed in the domain of the system. A performanceindex functional which comprises of modified energy functional of the system and theexpenditure of the actuators is also introduced. To minimize the performance indexfunctional calculus of variation is used and the necessary conditions for the optimal-ity are derived in the form of Fredholm integral equations. A numerical example isillustrated to show the applicability of the proposed technique. [1] [2] [3] [4] [5].

Bibliography

[1] Oniszczuk, Z. ”Free transverse vibrations of elastically connected simply sup-ported double-beam complex system,” Journal of sound and vibration, vol. 232,no. 2, pp. 387-403, 2000.

[2] Z. Oniszczuk, “Free transverse vibrations of an elastically connected complexbeam–string system,” Journal of Sound and Vibration, vol. 254, no. 4, pp.703–715, 2002.

[3] F. Cheli and G. Diana, “Vibrations in Continuous Systems,” in Advanced Dy-namics of Mechanical Systems. Springer International Publishing, 2015, pp.241–309.

[4] I. Kucuk and I. Sadek, “Optimal control of an elastically connected rectangularplate–membrane system,” Journal of Computational and Applied Mathematics,vol. 180, no. 2, pp. 345–363, 2005.

[5] I. Kucuk and I. Sadek, “Active vibration control of an elastically connecteddouble-string continuous system,” Journal of the Franklin Institute, vol. 344,no. 5, pp. 684–697, 2007.

25

Well-posedness of the fourth-order Schrodinger equation onthe half-line in fractional Sobolev spacesAugust 9th

15:50-16:15Turker Ozsarı, Nermin Yolcu*

Izmir Institute of Technology

In this talk, we consider the initial boundary value problem for the fourth orderSchrodinger equation on the right half-line in L2−based fractional Sobolev spaces:

iqt + ∆2q = 0, (x, t) ∈ R+ × (0, T ),

q(x, 0) = q0 ∈ Hs(R+),

q(0, t) = g0(t) ∈ H2s+3

8t (0, T ),

qx(0, t) = g1(t) ∈ H2s+1

8t (0, T ).

At the beginning, we briefly introduce the Fokas method (also known ”UnifiedTransform Method (UTM)”), which is a novel approach for obtaining representationformulas for the solutions of the initial-boundary value problems posed on the half-line. The representation formula obtained via UTM is then analyzed by using theFourier theory to deduce sharp estimates on the space and time traces of the solutionfrom which the well-posedness follows.

26

Analytical solution of the Volterra-Fredholm singularintegro-differential August 9th

16:15-16:40Fernane Khaireddine*

University of 8 May 1945 Guelma

In this paper, we apply Taylor’s approximation and then transform the givennth-order weakly singular linear Volterra and Fredholm integro-differential equa-tions with into an ordinary linear differential equation. Some different examples areconsidered the results of these examples indicated that the procedure of transfor-mation method is simple and effective, and could provide an accurate approximatesolution or exact solution.

We consider the following nth-order linear-weakly singular Fredholm and Volterraintegro-differential equations at the form :

y(n)(t) +n−1∑m=0

βm(t)y(m)(t) = f(t) +

∫ b

a

y(p)(s)

|t− s|αds, a ≤ t ≤ b, 0 < α < 1 (2)

with initial conditions

y(a) = µ0, y′(a) = µ1, ......, y

(n−1)(a) = µn−1 (3)

y(n)(t) +n−1∑m=0

βm(t)y(m)(t) = f(t) +

∫ t

a

y(p)(s)

(t− s)αds, a ≤ t ≤ b, 0 < α < 1 (4)

with initial conditions

y(a) = µ0, y′(a) = µ1, ......, y

(n−1)(a) = µn−1 (5)

where µi(i = 1, ...., n − 1); are real constants, n; p are positive integers and0 ≤ p ≤ n, f(t); βk(t)(k = 0, 1, ..., n−1) are given functions, and y(t) is the solutionto be determined.

The objectives of this work is to develop a new approach to resolve the high-orderlinear weakly-singular Fredholm and Volterra integro-differential equations in onedimensional space.The proposed method is a direct method in which we remove thesingularity by using Taylor’s approximation and rewrite the weakly-singular integro-differential Fredholm and Volterra equation as either a linear differential equationthat can be solved using the analytical method or the known numerical methods.

Bibliography

[1] L. Bougoffa, R. C. Rach and A. Mennouni, An approximate method for solvinga class of weakly-singular Volterra integro-differential equations. Appl. Math.Comput. 217 (2011), no. 22, 8907–8913.

27

[2] H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integralequations by collocation methods. SIAM J. Numer. Anal. 27 (1990), no. 4, 987–1000.

[3] A. Chakrabarti and A. J. George, A formula for the solution of general Abelintegral equation. Appl. Math. Lett. 7 (1994), no. 2, 87–90.

[4] Z. Chen and Y; Lin, The exact solution of a linear integral equation with weaklysingular kernel. J. Math. Anal. Appl. 344 (2008), no. 2, 726–734.

[5] D. Conte and I. D. Prete, Fast collocation methods for Volterra integral equationsof convolution type. J. Comput. Appl. Math. 196 (2006), no. 2, 652–663.

[6] G. Ebadi, M. Y. Rahimi-Ardabili and S. Shahmorad, Numerical solution of thenonlinear Volterra integro-differential equations by the tau method. Appl. Math.Comput. 188 (2007), no. 2, 1580–1586.

[7] M. Ghasemi, M. Tavassoli Kajani and E. Babolian, Application of He’s ho-motopy perturbation method to nonlinear integro-differential equations. Appl.Math. Comput. 188 (2007), no. 1, 538–548.

[8] U. Lepik, Haar wavelet method for nonlinear integro-differential equations. Appl.Math. Comput. 176 (2006), no. 1, 324–333.

[9] X. Lv and S. Shi, The combined RKM and ADM for solving nonlinear weaklysingular Volterra integrodifferential equations. Abstr. Appl. Anal. 2012, Art. ID258067, 10 pp.

[10] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of nonlinearintegral equation. Appl. Math. Comput. 167 (2005), no. 2, 1119–1129.

28

Non-Homogeneus Continuous and Discrete GradientSystems: The Quasi-convex case August 9th

16:40-17:05Vahid Mohebbi*

Roi de Janeiro IMPA

In this paper, first we study the weak and strong convergence of solutions to thefollowing first order nonhomogeneous gradient system

−x′(t) = ∇φ(x(t)) + f(t), a.e on (0,∞)

x(0) = x0 ∈ H

to a critical point of φ where φ is C1 quasi-convex function on a real Hilbert spaceH with Argminφ 6= ∅. These results extend the results of [?] to nonhomogeneouscase. Then the discrete version of the above system by backward Euler discretizationhas been studied. Beside of the proof of the existence of the sequence given by thediscrete system, some results on the weak and strong convergence to the criticalpoint of φ are also proved. These results when φ pseudo-convex (therefore thecritical points are the same minimum points) may be applied in optimization forapproximation of a minimum point of φ.

Key words: Gradient system, quasi-convex, backward Euler discretization,weak convergence, strong convergence.

Mathematics Subject Classification: 34D20; 37C75; 93D09

29

Existence and Continuation Theorems of Caputo TypeFractional Differential EquationsAugust 9th

17:05-17:30Shahzad Sarwar*

Shanghai University

In this talk, we consider the general fractional differential equation (FDE) withCaputo derivative and study the existence and continuation of solution. Firstly, weconstruct a new local existence theorem. Then we extend the continuation theoremsfor ODEs to those of general FDEs. Besides, several global existence results for FDEare constructed.

30

Global uniqueness and stability of an inverse problem forthe Schrodinger equation on a Riemannian manifold via one

boundary measurement August 10th9:10-10:00

Roberto Triggiani*

University of Memphis

We consider a mixed problem for the Schrodinger equation on a finite dimen-sional Riemannian manifold with magnetic and electric potential coefficients andnon-homogeneous Dirichlet boundary term. The goal is the nonlinear problem ofthe recovery of the electric potential coefficient by means of only one boundarymeasurement on an explicitly identified portion of the boundary. We obtain globaluniqueness of the recovery and Lipschitz stability of the recovery on an arbitrar-ily short time-interval exclusively in terms of the data of the problem. The normsinvolved are optimal. The key ingredients are:

(i) sharp Carleman-type estimates for the Schrodinger equation at the H1-level,on a Riemannian manifold and consequent continuous observability estimates at theH1-level (Triggiani-Xu, AMS,2007);

(ii) related continuous observability estimates at the L2-level (Lasiecka-Triggiani-Zhang; and Triggiani 2004)

(iii) optimal interior and boundary regularity of the direct mixed problem (Lasiecka-Triggiani, 1991) plus a new boundary trace result in the present contributionhShouhong Wang.

31

Error analysis of a semi-discrete method for nonlocalnonlinear wave equationsAugust 10th

10:00-10:50Husnu Ata Erbay*, Saadet Erbay, Albert Erkip

Ozyegin University

In this study we present a semi-discrete numerical method for solving the initial-value problem of the nonlocal nonlinear wave equation [1]

utt = (β ∗ f(u))xx .

Here the symbol ∗ is used to denote the convolution operation in the spatialdomain

(β ∗ v)(x) =

∫Rβ(x− y)v(y)dy

and the kernel β is an even function with∫R β(x)dx = 1. The numerical method

is based on a uniform spatial discretization of the convolution integral. We provide arigorous convergence analysis and demonstrate the second-order accuracy in space.The main ingredient in the convergence analysis is the uniform estimate of thesecond-order derivative of the kernel function. To confirm our theoretical findings,we apply the scheme to two different situations: the propagation of a single-solitarywave and the finite-time blow-up of solutions.

Bibliography

[1] N. Duruk, H.A. Erbay and A. Erkip, Global existence and blow-up for a class ofnonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity 23 (2010),107–118.

32

Stability estimates for some inverse problems forultrahyperbolic Schrodinger equations August 10th

11:10-11:35Fikret Golgeleyen*, Ozlem Kaytmaz

Bulent Ecevit University

In this work, we obtain a global Carleman estimate for an ultrahyperbolic Schrodingerequation which arises in several applications [2, 7]. Based on the idea by Imanuvilovand Yamamoto [3], we prove Holder stability for the inverse problem of determininga coefficient or a source term in the ultrahyperbolic Schrodinger equation by somelateral boundary data. To our best knowledge, there is no result available in themathematical literature related to the inverse problems for these equations. As forthe classical Schrodinger equation, we refer to Baudouin and Puel [1], Lasiecka etal. [4], Mercado et al. [5], and Yuan and Yamamoto [6].

Bibliography

[1] Baudouin, L., and Puel, J. P. (2002). Uniqueness and stability in an inverseproblem for the Schrodinger equation. Inverse Problems, 18(6), 1537.

[2] Davey, A., and Stewartson, K. (1974). On three-dimensional packets of surfacewaves. In Proceedings of the Royal Society of London A: Mathematical, Physicaland Engineering Sciences, 338(1613), 101-110.

[3] Imanuvilov, O. Y., and Yamamoto, M. (2001). Global Lipschitz stability in aninverse hyperbolic problem by interior observations. Inverse Problems, 17(4),717-728.

[4] Lasiecka, I., Triggiani, R., and Zhang, X. (2004). Global uniqueness, observ-ability and stabilization of nonconservative Schrodinger equations via pointwiseCarleman estimates. Part I: H1(Ω)-estimates. Journal of Inverse and Ill-posedProblems, 12(1), 43-123.

[5] Mercado, A., Osses, A., and Rosier, L. (2008). Inverse problems for theSchrodinger equation via Carleman inequalities with degenerate weights. In-verse Problems, 24(1), 015017.

[6] Yuan, G., and Yamamoto, M. (2010). Carleman estimates for the Schrodingerequation and applications to an inverse problem and an observability inequality.Chinese Annals of Mathematics, Series B, 31(4), 555-578.

[7] Zakharov, V. E., and Kuznetsov, E. A. (1986). Multi-scale expansions in thetheory of systems integrable by the inverse scattering transform. Physica D:Nonlinear Phenomena, 18(1-3), 455-463.

33

Long-time behaviour of the strongly damped semilinearplate equation in RnAugust 10th

11:35-12:00Sema Yayla*

Hacettepe University

We investigate the initial-value problem for the semilinear plate equation con-taining localized strong damping, localized weak damping and nonlocal nonlinearity.We prove that if nonnegative damping coefficients are strictly positive almost every-where in the exterior of the some ball and the sum of these coefficients is positive a.e.in Rn then the semigroup generated by the considered problem possesses a globalattractor in H2 (Rn) × L2 (Rn). We also establish boundedness of this attractor inH3 (Rn)×H2 (Rn).

Bibliography

[1] A. Pazy, Semigroups of linear operators and applications to partial differentialequations, Springer, New York, 1983.

[2] T. Cazenave and A. Haraux, An introduction to semilinear evolution equations,Oxford University Press, New York, 1998.

[3] E. Feireisl, Asymptotic behavior and attractors for a semilinear damped waveequation with supercritical exponent, Proc. Roy. Soc. Edinburgh, 125 (1995)1051–1062.

[4] M. Conti, V. Pata and M. Squassina, Strongly damped wave equations on R3

with critical nonlinearities, Commun. Appl. Anal., 9 (2005) 161–176.

[5] Z. Yang and P. Ding, Longtime dynamics of the Kirchhoff equation with strongdamping and critical nonlinearity on RN , J. Math. Anal. Appl., 434 (2016)1826-1851.

[6] J.M. Ball, Global attractors for damped semilinear wave equations, DiscreteContin. Dyn. Syst., 10 (2004) 31–52.

[7] Z. Arat, A. Khanmamedov and S. Simsek, Global attractors for the plate equa-tion with nonlocal nonlinearity in unbounded domains, Dyn. Partial Differ. Equ.,11 (2014) 361–379.

[8] A. Khanmamedov and S. Simsek, Existence of the global attractor for the plateequation with nonlocal nonlinearity in Rn, Discrete Contin. Dyn. Syst. Ser. B,21 (2016) 151–172.

[9] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations:Well-posednessand long-time dynamics, Springer, New York, 2010.

[10] J. Simon, Compact sets in the space Lp(0, T; B), Annali Mat. Pura Appl., 146(1987) 65-96.

34

[11] A.Kh. Khanmamedov, Global attractors for von Karman equations with non-linear interior dissipation, J. Math. Anal. Appl., 318 (2006) 92–101.

[12] A.Kh. Khanmamedov, Global attractors for 2-D wave equations with displace-ment dependent damping, Math. Methods Appl. Sci., 33 (2010) 177-187.

35

Small Data Global Existence and Exponential Decay ofSolutions to a Free Boundary Fluid-Structure Interaction

System with DampingAugust 10th13:00-13:50

Mihaela Ignatova, Igor Kukavica, Irena Lasiecka, Amjad Tuffaha*

American University of Sharjah

We consider a coupled system of PDEs consisting of an incompressible Navier-Stokes equation and a linear damped wave equation coupled through transmissionboundary conditions at a moving interface. This is a well-established free mov-ing boundary model describing fluid-elasticity interaction and its local-in-time well-posedness has been treated in several works in the literature [CS1, CS2, KT1, KT2,RV]. In an earlier work, we establish a similar small data global existence resultwhen the system is subject to both boundary and interior damping [IKLT1]. Inthis work, we establish global-in-time solutions for the system given small initialsmooth data and subject only to interior damping but no boundary dissipation, andwe show exponential decay of the solution in some Sobolev norm.

We first prove that any smooth solution is global and exponentially decayingif the initial data is small in an appropriate Sobolev norm. The main task is toestablish a priori superlinear estimates which guarantee control of the time integralof the wave potential energy. This is achieved using special multipliers and theequipartition energy technique for the wave equation. The second part of the proofconcerns the construction of global smooth solutions in a higher norm, to which thefirst result can be applied in order to conclude exponential decay in the lower Sobolevnorm. Here, we appeal to maximal regularity of parabolic equations [PS] especiallycrafted for Stokes equations with non-homogeneous divergence and Neumann typeboundary conditions [MZ1, MZ2], in addition to sharp trace regularity of solutionsto the wave equation [LLT]. This way we are able to construct solutions to the linearsystem (with given smooth variable coefficients in the Stokes equation). To addressthe nonlinear system where the variable coefficients in the Lagrangian formulationare also unknown, we use a fixed point iteration scheme to construct global solutionsto the full system.

Bibliography

[CS1] D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompress-ible viscous fluid, Arch. Ration. Mech. Anal. 176 (2005), no. 1, 25–102.

[CS2] D. Coutand and S. Shkoller, The interaction between quasilinear elastody-namics and the Navier-Stokes equations, Arch. Ration. Mech. Anal. 179 (2006),no. 3, 303–352.

[IKLT1] M. Ignatova, I. Kukavica, I. Lasiecka, and A. Tuffaha, On well-posednessand small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity 27 (2014), no. 3, 467–499.

[IKLT2] M. Ignatova, I. Kukavica, I. Lasiecka, and A. Tuffaha, Small data globalexistence for a fluid-structure model, Nonlinearity 30 (2017), 848–898.

36

Nonlinearity 30 (2017) 848–898

[KT1] I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction freeboundary problem, Discrete Contin. Dyn. Syst. 32 (2012), no. 4, 1355–1389.

[KT2] I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problemof fluid-structure interaction, Indiana Univ. Math. J. 61 (2012), no. 5, 1817–

[LLT] I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary valueproblems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65(1986), no. 2, 149–192.

[MZ1] P.B. Mucha and W.M. Zajaczkowski, On the existence for the Cauchy–Neumann problem for the Stokes system in the Lp-framework, Studia Math. 143(2000), 75–101.

[MZ2] P.B. Mucha and W.M. Zajaczkowski, On local existence of solutions of freeboundary problem for incompressible viscous self-gravitating fluid motion, Appli-cationes Mathematicae 27 (2000), no. 3, 319–333.

[PS] J. Pruss and G. Simonett, On the two-phase Navier-Stokes equations with sur-face tension, Interfaces Free Bound. 12 (2010), no. 3, 311–345.

[RV] J.-P. Raymond and M. Vanninathan, A fluid-structure model coupling theNavier-Stokes equations and the Lame system, J. Math. Pures Appl. (9) 102(2014), no. 3, 546–596.

37

Complex Ginzburg-Landau equations with dynamicboundary conditionsAugust 10th

13:50-14:40Wellington Jose Correa, Turker Ozsarı*

Izmir Institute of Technology

The initial-dynamic boundary value problem (idbvp) for the complex Ginzburg-Landau equation (CGLE) on bounded domains of RN is studied by converting thegiven mathematical model into a Wentzell initial-boundary value problem (ibvp).First, the corresponding linear homogeneous idbvp is considered. Secondly, theforced linear idbvp with both interior and boundary forcings is studied. Then, thenonlinear idbvp with Lipschitz nonlinearity in the interior and monotone nonlin-earity on the boundary is analyzed. The local well-posedness of the idbvp for theCGLE with power type nonlinearities is obtained via a contraction mapping argu-ment. Global well-posedness for strong solutions is shown. Global existence anduniqueness of weak solutions are proven. Smoothing effect of the corresponding evo-lution operator is proved. This helps to get better well-posedness results than theknown results on idbvp for nonlinear Schrodinger equations (NLS). An interestingresult is proving that solutions of NLS subject to dynamic boundary conditions canbe obtained as inviscid limits of the solutions of the CGLE subject to same type ofboundary conditions. Finally, long time behaviour of solutions is characterized andexponential decay rates are obtained at the energy level by using control theoretictools.

Turker Ozsarı’s research was supported by TUBITAK 3501 Career Grant 115F055

38

The Stability of an IMEX (Implicit-Explicit) Method forSimplified MHD (MagnetoHydroDynamics) Equations August 10th

15:00-15:25Simge Kacar Eroglu*, Gamze Yuksel

Mugla Sitki Kocman University

Magnetohydrodynamics (MHD) studies consider the dynamics of electrically con-ducting fluids. MHD are described by a set of equations,which are a combination ofthe Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electro-magnetism [1]. In most terrestrial applications, MHD flows occur at low magneticReynolds numbers. In this study, we apply the finite element method to time-dependent MHD flows with implicit-explicit (IMEX) method to discretization atlow magnetic Reynolds number. We introduce an imex (implicit-explicit) methodfor time discretization and also finite element method for space discretization [2, 3].Finally, the stability of the present method is presented comprehensively.

Key words: Magnetohydrodynamics, Imex methods, Finite Element Method

Bibliography

[1] P.A. Davidson. An Introduction to Magnetohydrodynamics. Cambridge Univer-sity Press, United Kingdom, 2001.

[2] C. Trenchea, Unconditional stability of a partitioned IMEX method for magne-tohydrodynamics flows, App. Math. Letters 27 (2014), 97-100.

[3] W. Layton, H. Tran, C. Trenchea, Numerical Analysis of Two Partitioned Meth-ods for Uncoupling Evolutionary MHD Flows, Numer. Meth. Part. D. E. 30(2014), no. 4, 1083-1102.

39

Inverse Source Problem for Benjamin-Bona-Mahony-BurgerEquationAugust 10th

15:25-15:50Serap Gumus*

Koc University

We study an inverse source problem for Benjamin-Bona-Mahony-Burger Equa-tion. Our aim is to prove existence, uniqueness of solution, continuous dependenceof solutions on initial data and asymptotic behavior of solutions as t→∞.

40

Boosting the decay of solutions of the Korteweg-deVries-Burgers equation to a predetermined rate from the

boundary August 10th15:50-16:15

Eda Arabacı*, Turker Ozsarı

Izmir Institute of Technology

We extend some recent results on the boundary feedback controllability of theKorteweg-de Vries equation (KdV) to the Korteweg-de Vries-Burgers equation (KdVB)which is posed on a bounded domain. In the first part of the talk, we discuss thefact that all sufficiently small solutions can be steered to zero at ”any desired” ex-ponential rate by means of a suitably constructed boundary feedback controller. Inthe second part, an observer is proposed when a type of boundary measurement isavailable while there is no full access to the medium.

41

TBAAugust 10th16:15-17:05

Varga Kalantarov

Koc University

To be announced...

42

Author Index

Abdelhamid Laouar*, 15Albert Erkip, 19Alp Eden*, 23Amjad Tuffaha*, 36Anna Kostianko*, 21Aseel Farhat, 18Attila Askar*, 9

Chongsheng Cao, 18Claude Bardos, 4

Eda Arabacı*, 41Edriss S. Titi*, 18Emre Kara*, 24

Fernane Khaireddine*, 27Fikret Golgeleyen*, 33

Gamze Yuksel, 39Gulzat Nalzhupbayeva*, 12

Husnu Ata Erbay, 32

Igor Kukavica, 36Irena Lasiecka, 36

Jamila Kalantarova*, 14

Melih Cınar*, 25

Meryem Kaya, 13Mihaela Ignatova, 36Mohammed Derhab, 16Mohammed Nehari*, 16Muharrem Ozluk*, 13Mustafa Taylan Sengul*, 20

Nader Masmoudi*, 8Nermin Yolcu*, 26Niyaz Tokmagambetov*, 6

Oktay Pashaev*, 5Orestiz Vantzos*, 7

Puskar R. Pokhrel*, 17

Roberto Triggiani*, 31

Sema Yayla*, 34Serap Gumus*, 40Sergey Zelik*, 22Shahzad Sarwar*, 30Shouhong Wang, 20Simge Kacar Eroglu*, 39

Turker Ozsarı, 14, 26, 38, 41

Vahid Mohebbi*, 29Varga Kalatarov, 42

43