Stability of FlowsS Friedlander, University of Illinois-Chicago,Chicago, IL, USA 2006 Elsevier Ltd. All rights reserved.IntroductionThis article gives a brief discussion of a topic withan enormous literature, namely the stability/instabil-ity of fluid flows. Following the seminal observa-tions and experiments of Reynolds in 1883, the issueof stability of a fluid flow became one of the centralproblems in fluid dynamics: stable flows are robustunder inevitable disturbances in the environment,while unstable flows may break up, sometimesrapidly. These possibilities were demonstrated in arelatively simple experiment where flow in a pipe isexamined at increasing speeds. As a dimensionlessparameter (now known as the Reynolds number)increases, the flow completely changes its naturefrom a stable flow to a completely different regimethat is irregular in space and time. Reynolds calledthis turbulence and observed that the transitionfrom the simple flow to the chaotic flow was causedby the phenomenon of instability.Even though the topic has been the subject ofintense study over more than a century, Reynoldsexperiment is still not fully explained by currenttheory. Although there is no rigorous proof ofstability of the simple flow (known as Poiseuilleflow in a circular pipe), analytical and numericalinvestigations of the equations suggest theoreticalstability for all Reynolds numbers. However, experi-ments show instability for sufficiently largeReynolds numbers. A plausible explanation for thisphenomenon is the instability of such flows withrespect to small but finite disturbances combinedwith their stability to infinitesimal disturbances.The issue of fluid stability, in contexts muchmore complex than the fundamental experiment ofReynolds, arises in a multitude of branches ofscience, including engineeering, physics, astrophy-sics, oceanography, and meteorology. It is farbeyond the scope of this short article to eventouch upon most of the extensive literature. In thebibliography we list just a few of the substantivebooks where classical results can be found(Chandrasekhar 1961, Drazin and Reid 1981,Gershuni and Zhukovitiskii 1976, Joseph 1976,Lin 1967, Swinney and Gollub 1985). Recentextensive bibliographies on mathematical aspectsof fluid instability are given in several articles in theHandbook of Mathematical Fluid Dynamics(Friedlander and Serre 2003) and the compendiumof articles on hydrodynamics and nonlinearinstabilities in Godreche and Maneville (1998).The Equations of MotionThe NavierStokes equations for the motion of anincompressible, constant density, viscous fluid are0q0t q r q 1j rP i r2q 1adiv q 0 1bwhere q(x, t) denotes the velocity vector, P(x, t) thepressure, and the constants j and i are the densityand kinematic viscosity, respectively. This system isconsidered in three (or sometimes two) spatialdimensions with a specified initial velocity fieldqx. 0 q0x 1cand physically appropriate boundary conditions: forexample, zero velocity on a rigid boundary, orperiodicity conditions for flow on a torus. Thisnonlinear system of partial differential equations(PDEs) has proved to be remarkably challenging,and in three dimensions the fundamental issues ofexistence and uniqueness of physically reasonablesolutions are still open problems.It is often useful to consider the NavierStokesequations in nondimensional form by scaling thevelocity and length by some intrinsic scale in theproblem, for example, in Reynolds experiment bythe mean speed U and the diameter of the pipe d.This leads to the nondimensional equations0q0t q r q rP 1R r2q 2adiv q 0 2bwhere the Reynolds number R isR Udi 3In many situations, the size of R has a crucialinfluence on stability. Roughly speaking, when R issmall the flow is very sluggish and likely to bestable. However, the effects of viscosity are actuallyvery complicated and not only is viscosity able tosmooth and stabilize fluid motions, sometimes itactually also destroys and destabilizes flows.The Euler equations, which predate the NavierStokes equations by many decades, neglect theeffects of viscosity and are obtained from [1a] bysetting the viscosity parameter i to zero. Since thisremoves the highest-derivative term from the equa-tions, the nature of the Euler equations is funda-mentally different from that of the NavierStokesequations and the limit of vanishing viscosity (orinfinite Reynolds number) is a very singular limit.Since all real fluids are at least very weakly viscous,it could be argued that only the the NavierStokesequations are physically relevant. However, manyimportant physical phenomena, such as turbulence,involve flows at very high Reynolds numbers (104orhigher). Hence, an understanding of turbulence islikely to involve the asymptotics of the NavierStokes equations as R!1. The first step towardsthe construction of such asymptotics is the study ofinviscid fluids governed by the Euler equations:0q0t q r q rP 4adiv q 0 4bStability issues for the Euler equations are in manyrespects distinct from those of the NavierStokesequations and in this article we will briefly touchupon stability results for both systems.Comments on Some ClassicalInstabilitiesTo illustrate the complexity of the structure ofinstabilities that can arise in the NavierStokesequations, we mention one classical example,namely the centrifugal instabilities called TaylorCouette instabilities. Consider a fluid between twoconcentric cylinders rotating with different angularvelocities. If the inner cylinder rotates sufficientlyfaster than the outer one, the centrifugal force isstronger on inside particles than outside particlesand a disturbance which exchanges the radialposition of particles is enhanced, that is, theconfiguration is unstable. As the angular velocityof the inner cylinder is increased above a certaincritical rate, the instability is manifested in a seriesof small toroidal (Taylor) vortices that fill the spacebetween the cylinders. There follows a hierarchy ofsuccessive instabilities: azimuthal traveling waves,twisting regimes, and quasiperiodic regimes untilchaotic solutions appear. Such a sequence ofbifurcations is a scenario for a transition toturbulence postulated by RuelleTakens. Detailsconcerning bifurcation theory and fluid behaviorcan be found in the book of Chossat and Iooss(1994).We note that phenomena of successive bifurca-tions connected with loss of stability, such asregimes of TaylorCouette instabilities, occur atmoderately large Reynolds numbers. Fully devel-oped turbulence is a phenomenon associated withvery high Reynolds numbers. These are parameterregimes basically inaccessible in current numericalinvestigations of the NavierStokes equations andturbulent models. The Euler equations lie at thelimit as R!1. It is an interesting observation thatresults at the limit of infinite Reynolds number aresometimes also applicable and consistent withexperiments for flows with only moderate Reynoldsnumber.There is a huge diversity of forces that couplewith fluid motion to produce instability. We willmerely mention a few of these which an interestedreader could pursue in consultation with texts listedin the Further readi ng sect ion and referenc estherein.1. The so-called Benard problem of convectiveinstability concerns a horizontal layer of fluidbetween parallel plates and subject to a tempera-ture gradient. The governing equations are theNavierStokes equation for a nonconstant den-sity fluid and the heat equation. In this problem,the critical parameter governing the onset ofinstability is called the Rayleigh number. Thepatterns that can develop as a result of instabilityare strongly influenced by the boundary condi-tions in the horizontal coordinates. With latticetype conditions, bifurcating solutions includerolls, rectangles, and hexagons. Convection rollsare themselves subject to secondary instabilitiesthat may break the translation symmetry anddeform the rolls into meandering shapes. Furtherrefinements of convective instabilities includedoubly diffusive convection, where the densitydepends on concentration as well as temperature.Competition between stabilizing diffusivity anddestabilizing diffusivity can lead to the so-calledsalt-finger instabilities.2. Of considerable interest in astrophysics andplasma physics are the instabilities that occur inelectrically conducting fluilds. Here the fluidequations are coupled with Maxwells equations.Much work has been done on the topic ofmagnetohydrodynamical (MHD) stability, whichwas developed to address various importantphysical issues such as thermonuclear fusion,stellar and planetary interiors, and dynamotheory. For example, dynamo theory addressesthe issue of how a magnetic field can begenerated and sustained by the motion of anelectrically conducting fluid. In the simplestscenario, the fluid motion is assumed to be agiven divergence-free vector field and the study of2 Stability of Flowsthe instabilities that may occur in the evolutionof the magnetic field is called the kinematicdynamo problem. This gives rise to interestingproblems in dynamical systems and actually isclosely analogous to the topic of vorticitygeneration in the three-dimensional (3D) fluidequations in the absence of MHD effects.In the next section we discuss certain mathema-tical results that have been rigorously proved forparticular problems in the stability of fluid flows.We restrict our attention to the basic equations,that is, [2a] and [2b], [4a] and [4b], observing thateven in rather simple configurations there are stillmore open problems than precise rigorous results.The NavierStokes Equations:Mathematical Definitions ofStability/InstabilityInstability occurs when there is some disturbance ofthe internal or external forces acting on the fluidand, loosely speaking, the question of stability orinstability considers whether there exist disturbancesthat grow with time. There are many mathematicaldefinitions of stability of a solution to a PDE. Mostof these definitions are closely related but they maynot be equivalent. Because of the distinctly differentnature of the NavierStokes equations for a viscousfluid and the Euler equations for an inviscid fluid,we will adopt somewhat different precise definitionsof stability for the two systems of PDEs. Bothdefinitions are related to the concept known asLyapunov stability. A steady state described by avelocity field U0(x) is called Lyapunov stable ifevery state q(x, t) close to U0(x) at t =0 staysclose for all t 0. In mathematical terms, close-ness is defined by considering metrics in a normedspace X. While in finite-dimensional systems thechoice of norm is not significant because all Banachnorms are equivalent, in infinite-dimensional sys-tems, such as a fluid configuration, this choice iscrucial. The point was emphasized by Yudovich(1989) and it is a version of the definition ofstability given in this book that we will adopt inconnection with the parabolic NavierStokesequations.Definitions for a General NonlinearEvolution EquationConsider an evolution equation for u(x, t) whosephase space is a Banach space X:0u0t Lu Nu. uWe assume that if the initial value u(x, 0) 2 X isgiven, the future evolution u(x, t), t 0, of theequation is uniquely defined (at least for sufficientlysmall initial data). Without loss of generality, wecan assume that zero is a steady state.We define a version of Lyapunov (nonlinear)stability and its converse instability.Definition 1 Let (X, Z) be a pair of Banach spaces.The zero steady state is called (X, Z) nonlinearlystable if, no matter how small c 0, there existsc 0 so that u(x, 0) 2 X andkux. 0kZ < cimply the following two assertions:(i) there exists a global in time solution such thatu(x, t) 2 (([0, 1); X);(ii) ku(x, t)kZ < c for a.e. t 2 [0, 1).The zero state is called nonlinearly unstable if eitherof the above assertions is violated. We note thatunder this strong definition of stability, loss ofexistence of a solution is a particular case ofinstability. The concept of existence that we willinvoke in considering the NavierStokes equations isthe existence of mild solutions introduced by Katoand Fujita (1962). Local-in-time existence of mildsolutions is known in X=Lqfor q ! n, where ndenotes the space dimension. (Lqdenotes the usualLebesque space).We now state two theorems for the NavierStokesequations [2a] and [2b]. The theorems are valid in anyspace dimension n and in finite or infinite domains. Ofcourse, the most physically relevant cases are n=3 or2. Both theorems relate properties of the spectrum ofthe linearized NavierStokes equations to stability orinstability of the full nonlinear system. LetU0(x), P0(x) be a steady state flow:U0 rU0 rP01Rr2U01RF 5ar U0 0 5bwhere U0 2 C1 vanishes on the boundary of thedomain D and F is a suitable external force. Wewrite [2a] and [2b] in perturbation form asqx. t U0x ux. t 6where0u0t LNSu Nu. u 7ar u 0 7bStability of Flows 3withLNSu U0 ru u rU01Rr2u rP1 8Nu. u u ru rP2 9Here P1 and P2 are, respectively, the portions of thepressure required to ensure that LNSu and N(u, u)remain divergence free. The operators LNS and N acton the space of divergence-free vector-valued func-tions in the closure of the Sobolev space Ws, pthatvanish on the boundary of D.We note that the spectrum of the elliptic linearoperator LNS with appropriate boundary conditionsin a bounded domain is purely discrete: that is, itconsists of a countable number of eigenvalues offinite multiplicity with the sole limit point being atinfinity.Theorem 2 (Nonlinear instability). Let 1 < p < 1be arbitrary. Suppose that the operator LNS over Lphas spectrum in the right half of the complex plane.Then the flow U0(x) is (Lq, Lp) nonlinearly unstablefor any q max(p, n).Theorem 3 (Asymptotic Lyapunov stability). Letq n be arbitrary. Assume that the operator LNSover Lqhas spectrum confined to the left half of thecomplex plane. Then the flow U0(x) is (Lq, Lq)nonlinearly stable.A recent proof of these theorems is given inFriedlander et al. (2006) using a bootstrap typeargument. In Theorem 2, the space Lq, q n, is usedas an auxiliary space inwhich the norm of thenonlinear term is controlled, while the final instabil-ity result is proved in Lpfor p 2 (1, 1). We notethat this includes the most physically relevant caseof instability in the L2energy norm. An earlier proofof the theorems under the restriction p ! n wasgiven by Yudovich (1989).To apply Theorem 2 or 3 to conclude nonlinearinstability or stability of a given flow U0, it isnecessary to have information concerning the spec-trum of the linear operator LNS. Obtaining suchinformation has been the goal of much of theliterature concerning fluid stability (see the biblio-graphy and the references therein). However, exceptin the case of some relatively simple flows, theeigenvalues of LNS have not yet been calculatedexplicitly. Perhaps the example that is the mosttractable is plane parallel shear flows. Here theeigenvalue problem is governed by an ordinarydifferential equation (ODE) known as the OrrSommerfeld equation, which has been the subject ofextensive analytical and numerical investigations.Consider the parallel flow U0=(U(z), 0, 0) in thestrip 1 z 1. For disturbances of the formcz eik1xk2ye`t10the eigenvalue ` is determined by the followingequation with k2=k21k22:U i`k d2dz2k2" #c U00 c 1ikRd2dz2k2" #c 11with boundary conditions c=0 at z =1. We notethat the discreteness of the spectrum is preserved ifperiodicity conditions are imposed in the (x, y)plane.The complexity of the spectral problem [11] isapparent even for the simple case U(z) =1 z2(known as plane Poiseuille flow). Unstable eigenva-lues exist but only in certain regions of (k, R)parameter space. There is a critical Reynolds number,Rc=5772, below which Re ` < 0 for all wavenumbers k. For R Rc, instability occurs in a bandof wave numbers and the thickness of this bandshrinks to zero as R!1 (i.e., the inviscid limit).Hence, Poiseuille flow with R < Rc can be consideredas an example where the stability Theorem 3 can beapplied, that is, the flow is nonlinearly stable toinfinitesimal disturbances. However, extremely care-ful experiments are needed to obtain agreement withthe theoretical value of Rc=5772. Rather it is moreusual in an experiment with R $ 2000 that the flowexhibits instability in the form of streamwise streaksthat appear near the walls. These structures do notlook like traveling waves of the form given byexpression [10], rather they are finite-amplitudeeffects of nonmodal growth. Such linear growth ofdisturbances, along with energy growth and pseudos-pectra have recently been investigated extensively.An example where Theorem 3, proving nonlinearinstability, can be applied is the so-calledKolmogorov flow. This is also a shear flow with thespectral problem for the linearized operator given byeqn [11]. In this example, the profile is oscillatory in zwith U(z) = sin mz. In an elegant paper, Meshalkinand Sinai (1961) used continued fractions to provethe existence of a real unstable positive eigenvalue. Itis interesting, and in some sense surprising, that theparticular case of sinusoidal profiles leads to anonconstant-coefficient eigenvalue problem, whereit is possible to construct in explicit form thetranscendental characteristic equation that relatesthe eigenvalues ` and the wave numbers. Usually,4 Stability of Flowsthis can be done only for constant-coefficient equa-tions. In the case U(z) = sin mz, a Fourier seriesrepresentation for the eigenfunctions leads to atridiagonal infinite matrix for the algebraic systemsatisfied by the Fourier coefficients. This is amenableto examination using continued fractions. Analysis ofthe characteristic equation shows that there exist realeigenvalues ` 0 provided R is larger than somecritical value for each wave number k with k2< m2.The Euler Equation: Linear andNonlinear Stability/InstabilityWe conclude this brief article with some discussionof instabilities in the inviscid Euler equations whoseexistence is likely to be important as a trigger forthe development of instabilities in high-Reynolds-number viscous flows. As we mentioned, the Eulerequations are very different from the NavierStokesequations in their mathematical structure. TheEuler equations are degenerate and nonelliptic. Assuch, the spectrum of the linearized operator LE isnot amenable to standard spectral theory of ellipticoperators. For example, unlike the NavierStokesoperator, the spectrum of LE is not purely discreteeven in bounded domains. To define LE we considera steady Euler flow {U0(x), P0(x)}, whereU0 rU0 rP0 12ar U0 0 12bWe assume that U0 2 C1. For the Euler equations,appropriate boundary conditions include zero nor-mal component of U0 on a rigid boundary, orperiodicity conditions (i.e., flow on a torus) orsuitable decay at infinity in an unbounded domain.The theorems that we will be describing have beenproved mainly in the cases of the second and thirdconditions stated above. There are many classes ofvector fields U0(x), in two and three dimensions,that satisfy [12a] and [12b]. We write [4a] and [4b]in perturbation form asqx. t U0x ux. t 13with0u0t LEu Nu. u 14ar u 0 14bHereLEu U0 r u u rU0rP1 15Nu. u u r u rP2 16Linear (spectral) instability of a steady Euler flowU0(x) concerns the structure of the spectrum of LE.Assuming U0 2 C1(Tn), the linear equation0u0t LEu. r u 0 17defines a strongly continuous group in every Sobolevspace Ws, pwith generator LE. We denote this groupby exp{LEt}. For the issue of spectral instability ofthe Euler equation it proves useful to study not onlythe spectrum of LE but also the spectrum of theevolution operator exp {LEt}. This permits thedevelopment of an explicit formula for the growthrate of a small perturbation due to the essential (orcontinuous) spectrum. It was proved by Vishik(1996) that a quantity , refered to as a fluidLyapunov exponent gives the maximum growthrate of the essential spectrum of exp{LEt}. Thisquantity is obtained by computing the exponentialgrowth rate of a certain vector that satisfies aspecific system of ODEs over the trajectories of theflow U0(x). This proves to be an effective mechan-ism for detecting instabilities in the essentialspectrum which result due to high-spatial-frequencyperturbations. For example, for this reason any flowU0(x) with a hyperbolic fixed point is linearlyunstable with growth in the sense of the L2-norm.In two dimensions, is equal to the maximalclassical Lyapunov exponent (i.e., the exponentialgrowth of a tangent vector over the ODE _ x=U0(x)).In three dimensions, the existence of a nonzeroclassical Lyapunov exponent implies that 0.However, in three dimensions there are also exam-ples where the classical Lyapunov exponent is zeroand yet 0. We note that the delicate issue of theunstable essential spectrum is strongly dependent onthe function space for the perturbations and that ,for a given U0, will vary with this function space.More details and examples of instabilities in theessential spectrum can be found in references in thebibliography.In contrast with instabilities in the essentialspectrum, the existence of discrete unstable eigenva-lues is independent of the norm in which growth ismeasured. From this point of view, such instabilitiescan be considered as strong. However, for mostflows U0(x) we do not know the existence of suchunstable eigenvalues. For fully 3D flows there are noexamples, to our knowledge, where such unstableeigenvalues have been proved to exist for flows withstandard metrics. The case that has received themost attention in the literature is the relativelysimple case of plane parallel shear flow. Theeigenvalue problem is governed by the RayleighStability of Flows 5equation (which is the inviscid version of the OrrSommerfeld equation [11]):U i`k d2dz2k2" #c U00c 0c 0 at z 1 18The celebrated Rayleigh stability criterion says thata sufficient condition for the eigenvalues ` to bepure imaginary is the absence of an inflection pointin the shear profile U(z). It is more difficult to provethe converse; however, there have been severalrecent results that show that oscillating profilesindeed produce unstable eigenvalues. For example, ifU(z) = sin mz the continued fraction proof ofMeshalkin and Sinai can be adapted to exhibit thefull unstable spectrum for [18]. We note the fluidLyapunov exponent is zero for all shear flows;thus the only way the unstable spectrum can benonempty for shear flows is via discrete unstableeigenvalues.As we have discussed, it is possible to show thatmany classes of steady Euler flows are linearlyunstable, either due to a nonempty unstable essentialspectrum (i.e., cases where 0) or due to unstableeigenvalues or possibly for both reasons. It is naturalto ask what this means about the stability/instabilityof the full nonlinear Euler equations [14][16]. Theissue of nonlinear stability is complex and there areseveral natural precise definitions of nonlinearstability and its converse instability.One definition is to consider nonlinear stabilityin the energy norm L2and the enstrophy norm H1,which are natural function spaces to measuregrowth of disturbances but are not correct spacesfor the Euler equations in terms of proven proper-ties of existence and uniqueness of solutions to thenonlinear equation. Falling under this definition isthe most frequently employed method to provenonlinear stability, which is an elegant techniquedeveloped by Arnold (cf. Arnold and Khesin(1998) and references therein). This is based onthe existence of the so-called energy-Casimirs. Thevorticity curl q is transported by the motion ofthe fluid so that at time t it is obtained from thevorticity at time t =0 by a volume-preservingdiffeomorphism. In the terminology of Arnold,the velocity fields obtained in this manner at anytwo times are called isovortical. For a given fieldU0(x), the class of isovortical fields is an infinite-dimensional manifold M, which is the orbit of thegroup of volume-preserving diffeomorphisms in thespace of divergence-free vector fields. The steadyflows are exactly the critical points of the energyfunctional E restricted to M. If a critical point is astrict local maximum or minimum of E, then thesteady flow is nonlinearly stable in the space J1 ofdivergence-free vectors u(x, t) (satisfying the bound-ary conditions) that have finite norm,kukJ1 kukL2 kcurl ukL2 19This theory can be applied, for example, to showthat any shear flow with no inflection points in theprofile U(z) is nonlinearly unstable in the functionspace J1, that is, the classical Rayleigh criterionimplies not only spectral stability but also nonlinearstability.We note that Arnolds stability method cannot beapplied to the Euler equations in three dimensionsbecause the second variation of the energy definedon the tangent space to M is never definite at acritical point U0(x). This result is suggestive, butdoes not prove, that most Euler flows in threedimensions are nonlinearly unstable in the Arnoldsense. To quote Arnold, in the context of the Eulerequations there appear to be an infinitely greatnumber of unstable configurations.In recent years, there have been a number ofresults concerning nonlinear instability for theEuler equation. Most of these results prove non-linear instability under certain assumptions on thestructure of the spectrum of the linearized Euleroperator. To date, none of the approaches provethe definitive result that in general linear instabilityimplies nonlinear instability. As we have remarked,this is a much more delicate issue for Euler than forNavierStokes because of the existence, for ageneric Euler flow, of a nonempty essentialunstable spectrum. To give a flavor of the mathe-matical treatment of nonlinear instability for theEuler equations, we present one recent result andrefer the interested reader to articles listed in theFurther reading section for f urther results anddiscussions.In the context of Euler equations in two dimen-sions, we adopt the following definition of Lyapu-nov stability.Definition 4 An equilibrium solution U0(x) iscalled Lyapunov stable if for every 0 there existsc 0 so that for any divergence-free vector u(x, 0) 2W1s, p, s 2p, such that ku(x, 0)kL2 < c the uniquesolution u(x, t) to [14][16] satisfieskux. tkL2 < for t 2 0. 1We note that we require the initial value u(x, 0) tobe in the Sobolev space W1s, p, s p2, since it isknown that the 2D Euler equations are globally intime well posed in this function space.6 Stability of FlowsDefinition 5 Any steady flow U0(x) for which theconditions of Definition 4 are violated is callednonlinearly unstable in L2.Observe that the open issues (in three dimensions)of nonuniqueness or nonexistence of solutions to[14][16] would, under Definition 5, be scenariosfor instability.Theorem 6 (Nonlinear instability for 2D Eulerflows). Let U0(x) 2 C1(T2) be satisfy [12]. Let be the maximal Lyapunov exponent to the ODE_ x=U0(x). Assume that there exists an eigenvalue `in the L2spectrum of the linear operator LE givenby [15] with Re ` . Then in the sense ofDefinition 5, U0(x) is Lyapunov unstable withrespect to growth in the L2-norm.The proof of this result is given in Vishik andFriedlander (2003) and uses a so-called bootstrapargument whose origins can be found in referencesin that article. We remark that the above result givesnonlinear instability with respect to growth of theenergy of a perturbation which seems to be aphysically reasonable measure of instability.In order to apply Theorem 6 to a specific 2D flowit is necessary to know that the linear operator LEhas an eigenvalue with Re ` . As we havediscussed, such knowledge is lacking for a genericflow U0(x). Once again, we turn to shear flows. Aswe noted =0 for shear flows, any shear profile forwhich unstable eigenvalues have been proved toexist provides an example of nonlinear instabilitywith respect to growth in the energy.We conclude with the observation that it istempting to speculate that, given the complexityof flows in three dimensions, most, if not all, suchinviscid flows are nonlinearly unstable. It is clearfrom the concept of the fluid Lyapunov exponentthat stretching in a flow is associated withinstabilities and there are more mechanisms forstretching in three, as opposed to two, dimensions.However, to date there are virtually no mathema-tical results for the nonlinear stability problem forfully 3D flows and many challenging issues remainentirely open.AcknowledgmentsThe author is very grateful to IHES and ENS-Cachan for their hospitality during the writing ofthis paper. She thanks Misha Vishik for muchhelpful advice.The work is partially supported by NSF grantDMS-0202767.See also: Compressible Flows: Mathematical Theory;Incompressible Euler Equations: Mathematical Theory;Magnetohydrodynamics; Newtonian Fluids andThermohydraulics; Non-Newtonian Fluids; TopologicalKnot Theory and Macroscopic Physics.Further ReadingArnold VI and Khesin B (1998) Topological Methods inHydrodynamics. New York: Springer.Chandrasekhar S (1961) Hydrodynamic and HydromagneticStability. Oxford: Oxford University Press.Chossat P and Iooss G (1994) The CouetteTaylor Problem.Berlin: Springer.Drazin PG and Reid WH (1981) Hydrodynamic Stability.Cambridge: Cambridge University Press.Friedlander S, Pavlovic N, and Shvydkoy R (2006) Nonlinearinstability for the NavierStokes equations (to appear inCommunications in Mathematical Physics).Friedlander S and Serre D (eds.) (2003) Handbook of Mathema-tical Fluid Dynamics, vol. 2. Amsterdam: Elsevier.Friedlander S and Yudovich VI (1999) Instabilities in fluidmotion. Notices of the American Mathematical Society 46:13581367.Gershuni GZ and Zhukovitiskii EM (1976) Convective Instabil-ity of Incompressible Fluids. Jerusalem: Keter PublishingHouse.Godreche C and Manneville P (eds.) (1998) Hydrodynamics andNonlinear Instabilities. Cambridge: Cambridge UniversityPress.Joseph DD (1976) Stability of Fluid Motions, 2 vols. Berlin:Springer.Kato T and Fujita H (1962) On the nonstationary NavierStokessystem. Rend. Sem. Mat. Univ. Padova 32: 243260.Lin CC (1967) The Theory of Hydrodynamic Stability.Cambridge: Cambridge University Press.Meshalkin LD and Sinai IaG (1961) Investigation of stability for asystem of equations describing the plane movement of anincompressible viscous liquid. App. Math. Mech. 25:17001705.Swinney H and Gollub L (eds.) (1985) Hydrodynamic Instabilitiesand Transition to Turbulence. New York: Springer.Vishik MM (1996) Spectrum of small oscillations of an ideal fluidand Lyapunov exponents. Journal de Mathe matiques Pures etApplique es 75: 531557.Vishik M and Friedlander S (2003) Nonlinear instability in2 dimensional ideal fluids: the case of a dominanteigenvalue. Communications in Mathematical Physics 243:261273.Yudovich VI (1989, US translation) Linearization Method inHydrodynamical Stability Theory, Transl. Math. Monog.vol. 74. Providence: American Mathematical Society.Stability of Flows 7Stability of MatterJ P Solovej, University of Copenhagen, Copenhagen,Denmark 2006 Elsevier Ltd. All rights reserved.IntroductionThe theorem on stability of matter is one of the mostcelebrated results in mathematical physics. It is oneof the rare cases where a result of such greatimportance to our understanding of the worldaround us appeared first in a completely rigorousformulation.Issues of stability are, of course, extremely impor-tant in physics. One of the major triumphs of thetheory of quantum mechanics is the explanation itgives of the stability of the hydrogen atom (and thecomplete description of its spectrum). Quantummechanics or, more precisely, the uncertainty princi-ple explains not only the stability of tiny microscopicobjects, but also the stability of gigantic stellarobjects such as white dwarfs. Chandrasekharsfamous theory on the stability of white dwarfsrequired, however, not only the usual uncertaintyprinciple, but also the Pauli exclusion principle forthe fermionic electrons.Whereas both the stability of atoms and thestability of white dwarfs were early triumphs ofquantum mechanics, it, surprisingly, took nearly40 years before the question of stability of everydaymacroscopic objects was even raised (Fisher andRuelle 1966). The rigorous answer to the questioncame shortly thereafter in what came to be knownas the theorem on stability of matter proved firstby Dyson and Lenard (1967).Both the stability of hydrogen and the stability ofwhite dwarfs simply mean that the total energy ofthe system cannot be arbitrarily negative. If therewere no such lower bound to the energy, one wouldhave a system from which it would be possible, inprinciple, to extract an infinite amount of energy.One often refers to this kind of stability as stabilityof the first kind.Stability of matter is somewhat different. Stabilityof the first kind for atoms generalizes, as noted later,to objects of macroscopic size. The question arisesas to how the lowest possible energy depends on thesize or, more precisely, on the (macroscopic) numberof particles in the object. Stability of matter in itsprecise mathematical formulation is the requirementthat the lowest possible energy depends at mostlinearly on the number of particles. Put differently,the lowest possible energy calculated per particlecannot be arbitrarily negative as the number ofparticles increases. This is often referred to asstability of the second kind. If stability of thesecond kind does not hold, one would be able toextract an arbitrarily large amount of energy byadding a single atomic particle to a sufficiently largemacroscopic object.A perhaps more intuitive notion of stability isrelated to the volume occupied by a macroscopicobject. More precisely, the volume of the object,when its total energy is close to the lowest possibleenergy, grows at least linearly in the number ofparticles. This volume dependence is a fairly simpleconsequence of stability of matter as formulatedabove.The first mention of stability of the second kindfor a charged system is perhaps by Onsager (1939),who studied a system of charged classical particleswith a hard core and proved the stability of thesecond kind. The proof of stability of matter byDyson and Lenard, which does not rely on any hard-core assumption, but rather on the properties offermionic quantum particles, used results fromOnsagers paper.The real relevance of the notion of stability of thesecond kind was first realized by Fisher and Ruelle(1966) in an attempt to understand the thermo-dynamic properties of matter and to give meaningto thermodynamic quantities such as the energydensity (energy per volume). Stability of matter is anecessary ingredient in explaining the existence ofthermodynamics, that is, that the energy pervolume has a well-defined limit as the volume andnumber of particles tend to infinity, with the ratio(i.e., the density of particles) kept fixed. Theexistence of this limit is, however, not just a simpleconsequence of stability of matter. The existence ofthe thermodynamic limit for ordinary chargedmatter was proved rigorously by Lieb and Lebowitz(1972) using the result on stability of matter as aninput.After the original proof of stability of matter byDyson and Lenard, several other proofs were given(see, e.g., reviews by Lieb (1976, 1990, 2004) fordetailed references). Lieb and Thirring (1975) inparticular presented an elegant and simple proofrelying on an uncertainty principle for fermions. Asexplained in a later section, the best mathematicalformulation of the usual uncertainty principle is interms of a Sobolev inequality. The method of Lieband Thirring is related to a Sobolev type inequalityfor antisymmetric functions. The LiebThirringinequality is discussed later. The proof by Dyson8 Stability of Matterand Lenard gave a very poor bound on the lowestpossible energy per particle. The proof by Lieb andThirring gave a much more realistic bound on thisquantity (see below). Two proofs of stability ofmatter will be sketched here. Both proofs rely on theLiebThirring inequality. The first proof described ismathematically simple to explain, whereas thesecond proof (LiebThirring) is based on theThomasFermi theory. It is mathematically some-what more involved but, from a physical point ofview, more intuitive.As in the case of white dwarfs, stability of matterrelies on the fermionic property of electrons. Dyson(1967) proved that the stability of the second kindfails if we ignore the Pauli exclusion principle. Inphysics textbooks, the importance of the Pauliexclusion principle for the stability of white dwarfsis often emphasized. Its importance for the stabilityof everything around us is usually ignored.As mentioned above the result on stability ofmatter appeared from the beginning as a completelyrigorously proved theorem. In contrast, the stabilityof white dwarfs was only derived rigorously by Lieband Thirring (1984) and Lieb and Yau (1987) over50 years after the original work of Chandrasekhar.The original formulation of stability of matter,which is given in the next section, dealt withcharged matter consisting of electrons and nucleiinteracting only through electrostatic interactionsand being described by nonrelativistic quantummechanics. Over the years, many generalizations ofstability of matter have been derived in order toinclude relativistic effects and electromagnetic inter-actions. Some of these generalizations will bediscussed in this article. A complete understandingof stability of matter in quantum electrodynamics(QED) does not exist as yet, which is intimatelyrelated to the fact that this theory still awaits amathematically satisfactory formulation.The Formulation of Stability of MatterConsider Knuclei with nuclear charges z1, . . . , zK 0at positions r1, . . . , rK 2 R3, and N electrons withcharges 1 (this amounts to a choice of units) atpositions x1, . . . , xN 2 R3. In order to discussstability, it turns out that one can consider thenuclei as fixed in space, whereas the electrons aredynamic. More precisely, this means that thekinetic energy of the nuclei is ignored. It isimportant to realize that if stability holds for staticnuclei, it also holds for dynamic nuclei. This issimply because the kinetic energy is positive, so thatthe effect of ignoring it is to lower the total energy.Since we consider only electrostatic interactions,the quantum Hamiltonian describing this system isHN XNi1Ti XKk1XNi1zkjxi rkjX1i Figure 7 Illustration of the dynamic adaption strategy inwavelet coefficient space.418 Wavelets: Application to Turbulenceshow the production of enstrophy and the concomi-tant dissipation of energy when the vortex dipolehits the wall.This computation illustrates the fact that theadaptive wavelet method allows an automatic gridrefinement, both in the boundary layers at thewall and also in shear layers which develop duringthe flow evolution far from the wall. Therewith,the number of grid points necessary for thecomputation is significantly reduced, and we con-jecture that the resulting compression rate willincrease with the Reynolds number.(a) (b)Figure 8 Dipole wall interaction at Re =1000. (a) Vorticity field, (b) corresponding centers of the active wavelets, at t =0.2, 0.4, 0.6,and 0.8 (from top to bottom).0.20.250.30.350.40.450.5501001502002503000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8E(t )Z(t )E(t )Z(t )tFigure 9 Time evolution of energy (solid line) and enstrophy(dashed line).Wavelets: Application to Turbulence 419AcknowledgmentsMarie Farge thankfully acknowledges Trinity Col-lege, Cambridge, UK, and the Centre Internationalde Rencontres Mathematiques (CIRM), Marseille,France, for hospitality while writing this paper.See also: Turbulence Theories; Viscous IncompressibleFluids: Mathematical Theory; Wavelets: Applications;Wavelets: Mathematical Theory.Further ReadingCohen A (2000) Wavelet methods in numerical analysis. In:Ciarlet PG and Lions JL (eds.) Handbook of NumericalAnalysis, vol. 7, Amsterdam: Elsevier.Cuypers Y, Maurel A, and Petitjeans P (2003) Physics ReviewLetters 91: 194502.Dahmen W (1997) Wavelets and multiscale methods for operatorequations. Acta Numerica 6: 55228.Daubechies I (1992) Ten Lectures on Wavelets. SIAM.Daubechies I, Grossmann A, and Meyer Y (1986) Journal ofMathematical Physics 27: 1271.Farge M (1992) Wavelet transforms and their applications toturbulence. Annual Reviews of Fluid Mechanics 24: 395457.Farge M and Rabreau G (1988) Comptes Rendus Hebdomadairesdes Seances de lAcademie des Sciences, Paris 2: 307.Farge M, Kevlahan N, Perrier V, and Goirand E (1996)Wavelets and turbulence. Proceedings of the IEEE 84(4):639669.Farge M, Kevlahan N, Perrier V, and Schneider K (1999)Turbulence analysis, modelling and computing using wavelets.In: van den Berg JC (ed.) Wavelets in Physics, pp. 117200.Cambridge: Cambridge University Press.Farge M and Schneider K (2002) Analysing and computingturbulent flows using wavelets. In: Lesieur M, Yaglom A, andDavid F (eds.) New trends in turbulence, Les Houches 2000,vol. 74, pp. 449503. Springer.Farge M, Schneider K, Pellegrino G, Wray AA, and Rogallo RS(2003) Physics of Fluids 15(10): 2886.Grossmann A and Morlet J (1984) SIAM J. Appl. Anal 15: 723.Lemarie P-G and Meyer Y (1986) Revista Matematica Ibero-americana 2: 1.Mallat S (1989) Transactions of the American MathematicalSociety 315: 69.Mallat S (1998) AWavelet Tour of Signal Processing. Academic Press.Schneider K, Farge M, and Kevlahan N (2004) Spatialintermittency in two-dimensional turbulence. In: TongringN and Penner RC (eds.) Woods Hole Mathematics, Perspec-tives in Mathematics and Physics, pp. 302328. WordScientific.http://wavelets.ens.fr other papers about wavelets and turbu-lence can be downloaded from this site.Wavelets: ApplicationsM Yamada, Kyoto University, Kyoto, Japan 2006 Elsevier Ltd. All rights reserved.IntroductionWavelet analysis was first developed in the early1980s in the field of seismic signal analysis in theform of an integral transform with a localized kernelfunction with continuous parameters of dilation andtranslation. When a seismic wave or its derivativehas a singular point, the integral transform has ascaling property with respect to the dilation para-meter; thus, this scaling behavior can be available tolocate the singular point. In the mid-1980s, theorthonormal smooth wavelet was first constructed,and later the construction method was generalizedand reformulated as multiresolution analysis(MRA). Since then, several kinds of wavelets havebeen proposed for various purposes, and the conceptof wavelet has been extended to new types of basisfunctions. In this sense, the most important effect ofwavelets may be that they have awakened deepinterest in bases employed in data analysis and dataprocessing. Wavelets are now widely used in variousfields of research; some of their applications arediscussed in this article.From the perspective of timefrequency analysis,the wavelet analysis may be regarded as a windowedFourier analysis with a variable window width,narrower for higher frequency. The wavelets cantherefore give information on the local frequencystructure of an event; they have been applied tovarious kinds of one-dimensional (1D) or multi-dimensional signals, for example, to identify anevent or to denoise or to sharpen the signal.1D wavelets (a,b)(x) are defined asa.bx 1jajp x ba where a( 60), b are real parameters and (x) is aspatially localized function called analyzing wave-let or mother wavelet. Wavelet analysis gives adecomposition of a function into a linear combina-tion of those wavelets, where a perfect reconstruc-tion requires the analyzing wavelet to satisfy somemathematical conditions.For the continuous wavelet transform (CWT),where the parameters (a, b) are continuous, theanalyzing wavelet (x)L2(R) has to satisfy theadmissibility condition420 Wavelets: Applicationsanalyzing wavelet (x)L2(R) has to satisfy theadmissibility conditionC Z 11j^.j2j.j d. < 1where (.) is the Fourier transform of (x):^. Z 11ei.xx dxThe admissibility condition is known to be equiva-lent to the condition that (x) has no zero-frequencycomponent, that is, (0) =0, under some mildcondition for the decay rate at infinity. Then theCWT and its inverse transform of a data functionf (x) 2 L2(R) is defined asTa. b 1CpZ 11a.bxf x dxf x 1CpZ 11Z 11Ta. ba.bx da dba2In the case of the discrete wavelet transform(DWT), the parameters (a, b) are taken discrete; atypical choice is a =12j, b=k2j, where j and k areintegers:j.kx 2j22jx kIn order that the wavelets {j,k(x) j j, k 2 Z} mayconstitute a complete orthonormal system in L2(R),the analyzing wavelet should satisfy more stringentconditions than the admissibility condition for theCWT, and is now constructed in the framework ofMRA. A data function is then decomposed by theDWT asf x X1j1cj.kj.kx. cj.k Z 11j.kxf x dxEven when the discrete wavelets do not constitutea complete orthnormal system, they often form awavelet frame if linear combinations of the waveletsare dense in L2(R) and if there are two constants A,B such that the inequalityAkf k2

Xj.kjhj.k. f ij2 Bkf k2holds for an arbitrary f (x) 2 L2(R). For the waveletframe {j,k}, there is a corresponding dual frame,{ j,k}, which permits the following expansion of f (x):f x Xj.khj.k. f i~j.kx Xj.kh~j.k. f ij.kxThe wavelet frame is also employed in severalapplications.From the prospect of applications, the CWTs arebetter adapted for the analysis of data functions,including the detection of singularities and patterns,while the DWTs are adapted to the data processing,including signal compression or denoising.Singularity Detection and MultifractalAnalysis of FunctionsSince its birth, the wavelet analysis has been appliedfor the detection of singularity of a data function.Let us define the Ho lder exponent h(x0) at x0 of afunction f (x) is defined here as the largest value ofthe exponent h such that there exists a polynomialPn(x) of degree n that satisfies for x in theneighborhood of x0:jf x Pnx x0j Ojx x0jhThe data function is not differentiable if h(x0) < 1,but if h(x0) 1 then it is differentiable and asingularity may arise in its higher derivatives. Thewavelet transform is applied to find the Ho lderexponent h(x0), because T(a, b) has an asymptoticbehavior T(a, b) =O(ah(x012))(a ! 0) if the ana-lyzing wavelet has N( h(x0)) vanishing moments,that is,Z 11xmx dx 0. m 2 Z. 0 m < NA commonly used analyzing wavelet for this purposemay be the N-time derivative of the Gaussianfunction (x) =dN(ex22)dxN. This method workswell to examine a single or some finite number ofsingular points of the data function.When the data function is a multifractal functionwith an infinite number of singular point of variousstrengths, the multifractal property of the datafunction is often characterized by the singularityspectrum D(h) which denotes the Hausdorff dimen-sion of the set of points where h(x) =h. Thesingularity spectrum is, however, difficult to obtaindirectly from the CWT, and the Legendre transfor-mation is introduced to bypass the difficulty.Fully developed 3D fluid turbulence may be atypical example of wavelet application to thesingularity detection. The Kolmogorov similaritylaw of fluid turbulence for the longitudinal velocityincrement u(r) e (u(x re) u(x)), where u(x)is the velocity field and e is a constant unit vector,Wavelets: Applications 421predicts a scaling property of the structure function;for r in the inertial subrange,hurpi $ r.p. .p p3where h i denotes the statistical mean. In reality,however, the scaling exponent .p measured inexperiments shows a systematic deviation from p/3,which is considered to be a reflection of intermit-tency, namely the spatial nonuniformity or multi-fractal property of active vortical motions inturbulence. For simplicity, let us consider thevelocity field on a linear section of the turbulencefield. According to the multifractal formalism, theturbulence velocity field has singularities of variousstrengths described by the singularity spectrumD(h), which is related to the scaling exponent .pthrough the Legendre transform, D(h) = infp(ph .p 1). This relation is often used to determine D(h)from the knowledge of .p (structure functionmethod). However, this method does not necessarilywork well because, for example, it does not capturethe singular points of the Ho lder exponent largerthan 1 and it is unstable for h < 0.These difficulties are not restricted to the turbu-lence research, but arise commonly when thestructure function is employed to determine thesingularity spectrum. In these problems, the CWTT(a, b) provides an alternative method. An inge-nious technique is to take only the modulus maximaof T(a, b) (for each of fixed a) to construct apartition functionZa. q Xl2Lmaxsupa.b02ljTa. b0j" #qwhere q 2 R, and Lmax denotes the set of all maximalines, each of which is a continuous curve for smallvalue of a, and there exists at least one maxima linetoward a singular point of the Ho lder exponenth(x0) < N. In the limit of a ! 0, defining theexponent t(q) as Z(a, q) $ at(q), one can obtain thesingularity spectrum through the Legendretransform:Dh infqq h 12 tq This method (wavelet-transform modulus-maxima(WTMM) method) is advantageous in that it worksalso for singularities of h 1 and h < 0. Severalsimple examples of multifractal functions have beensuccessfully analyzed by this method. For fluidturbulence, this method gives a singularity spectrumD(h) which has a peak value of $1 at h $13,consistently with Kolmogorov similarity law, buthas a convex shape around h =13 suggesting amultifractal property. For a fractal signal, we notethat the WTMM method enlightens the hierarchicalorganization of the singularities, in the branchingstructure of the WT skeleton defined by themaxima lines arrangement in the (a, b) half-plane.Though the above discussion also applies to theDWT, the detection of the Ho lder exponent h inexperimental situations is usually performed by theCWT, which has no restriction on possible values ofa, while the DWT is often employed for theoreticaldiscussions of singularity and multifractal structureof a function.Multiscale AnalysisWavelet transform expands a data function in thetimefrequency or the positionwavenumber space,which has twice the dimension of the original signal,and makes it easier to perform a multiscale analysisand to identify events involved in the signal. In thewavelet transform, as stated above, the time resolu-tion is higher at higher frequency, in contrast withthe windowed Fourier transform where the time andthe frequency resolutions are independent of fre-quency. Another advantage of wavelet is a widevariety of analyzing wavelet, which enables us tooptimize the wavelet according to the purpose ofdata analysis. Both the CWT and the DWT areavailable for these timefrequency or positionwavenumber analysis. However, the CWT hasproperties quite different from those of familiarorthonormal bases of discrete wavelets.Multidimensional CWTThe CWT can be formulated in an abstract way. Wecan regard G={(a, b) j a( 60), b 2 R} as an affinegroup on R with the group operation of(a, b)(a0, b0) =(aa0, ab0 b) associated with theinvariant measure dj=da dba2. The group G hasits unitary representation in the Hilbert spaceH=L2(R):Ua. bf x 1jajp f x ba and then we can consider the CWT can be constructedas a linear map W from L2(R) to L2(G; da dba2):W : f x 7!Ta. b 1Cp hUa. b. f iwhere h , i is the inner product of L2(R) with thecomplex conjugate taken at the first element, and422 Wavelets: Applications(x) is a unit vector (analyzing wavelet) satisfyingthe abstract admissibility conditionC ZGjhUa. b. ij2dj < 1This formulation is applicable also to a locallycompact group G and its unitary and squareintegrable representation in a Hilbert space H.Note that even the canonical coherent states areincluded in this framework by taking the WeylHeisenberg group and L2(R) for G and H,respectively. This abstract formulation allows usto extend the CWT to higher-dimensional Eucli-dean spaces and other manifolds: for example, 2Dsphere S2for geophysical application and 4Dmanifold of spacetime taking the Poincare groupinto consideration.In Rn, the CWT of f (x) 2 L2(Rn) and its inversetransform are given byTa. r. b 1CpZRna.r.bxf x dxf x 1CpZGTa. r. ba.r.bxda dr dban1where r 2 SO(n), b 2 Rn, dr is the normalized invar-iant measure of G=SO(n), and the wavelets aredefined as (a, r, b)(x) =(1an2)(r1(x b)a), withthe analyzing wavelet satisfying the admissibilityconditionC ZRnj^wjjwjn dw < 1Note that these wavelets are constructed not onlyby dilation and translation but also by rotationwhich therefore gives the possibility for directionalpattern detection in a data function. In the case of2D sphere S2, on the other hand, the dilationoperation should be reinterpreted in such a waythat at the North Pole, for example, it is the normaldilation in the tangent plane followed by lifting itto S2by the stereographic projection from theSouth Pole.Generally, the abstract map W thus defined isinjective and therefore reversal, but not surjective incontrast with the Fourier case. Actually in the case of1D CWT, T(a, b) is subject to an integral condition:Ta. b Z 11Z 11dadba2 Ka. b; a0. b0Ta0. b0Ka. b; a0. b0 Z 11a.bxa0.b0x dxwhich defines the range of the CWT, a subspaceof L2(R). Therefore, if one wants to modify T(a, b)by, for example, assigning its value as zero in someparameter region just as in a filter process, careshould be taken for the resultant T(a, b) to be in theimage of the CWT. The reason may be understoodintuitively by noticing that the wavelets (a,b)(x) arelinearly dependent on each other. The expression ofa data function by a linear combination of thewavelets is therefore not unique, and thus isredundant. The CWT gives only T(a, b) of theleast norm in L2(R2; dadba2). In physical inter-pretations of the CWT, however, this nonuniquenessis often ignored.Pattern DetectionEdge detection The edges of an object are often themost important components for pattern detection.The edge may be considered to consist of points ofsharp transition of image intensity. At the edge, themodulus of the gradient of the image f (x, y) isexpected to take a local maximum in the 1Ddirection perpendicular to the edge. Therefore, thelocal maxima of jrf (x, y)j may be the indicator ofthe edge. However, the image textures can also givesimilar sharp transitions of f (x, y), and one shouldtake into account the scale dependence whichdistinguishes between edges and textures. One ofthe practically possible ways for this purpose is touse dyadic wavelets mj (x, y) =2jm(2jx, 2jy) whichare generated from the two wavelets (1, 2) =(000x, 000y), where 0 is a localized function(multiscale edge detection method). The dyadicwavelet transform of the image f (x, y)Tmj b1. b2 hf x. y. mj x b1. y b2i. m 1. 2defines the multiscale edges as a set of pointsb=(b1, b2) where the modulus of the wavelet trans-form, j(T1j , T2j )j, takes a locally maximum value(WTMM) in a 1D neighborhood of b in thedirection of (T1j (b), T2j (b)). Scale dependence ofthe magnitude of the modulus maxima is related tothe Ho lder exponent of f (x, y) similarly to 1D case,and thus gives information to distinguish betweenthe edges and the textures.Inversely, the information of WTMM bj,p ={(b1,j,p, b2,j,p)} of multiscale edges can be made useof for an approximate reconstruction of the originalimage, although the perfect reconstruction cannot beexpected because of the noncompleteness of themodulus maxima wavelets. Assuming that{1j,p, 2j,p} ={1j (x bj,p), 2j (x bj,p)} constitutes aframe of the linear closed space generated byWavelets: Applications 423{1j,p, 2j,p}, an approximate image f is obtained byinverting the relationLf XmXj.phf . mj.pimj.p XmXj.pTmj bj.pmj.pusing, for example, a conjugate gradient algorithm,where a fast calculation is possible with a filter bankalgorithm for the dyadic wavelet (algorithm a`trous). This algorithm gives only the solution ofminimum norm among all possible solutions, but itis often satisfactory for practical purposes and thusis applicable also to data compression.Directional detection For oriented features such assegments or edges in images to be detected, adirectionally selective wavelet for the CWT is desired.A useful wavelet for this purpose is one that has theeffective support of its Fourier transform in a convexcone with apex at the origin in wave number space. Atypical example of the directional wavelet may be the2D Morlet wavelet:x expik0 x expjAxj2where k0 is the center of the support in Fourierspace, and A is a 2 2 matrix diag[c12, 1](c 1),where the admissibility condition for the CWT isapproximately satisfied for jk0j ! 5. Another exam-ple is the Cauchy wavelet which has the supportstrictly in a convex cone in wave number space.These wavelets have the directional selectivitywith preference to a slender object in a specificdirection. One of their applications is the analysis ofthe velocity field of fluid motion from an experi-mental data, where many tiny plastic balls distrib-uted in fluid give a lot of line segments in a picturetaken with a short exposure. The directional waveletanalysis of the picture classifies the line segmentsaccording to their directions, indicating the direc-tions of fluid velocity. Another example may be awave-field analysis where many waves in differentdirections are superimposed; the directional waveletsallow one to decompose the wave field into thecomponent waves. Directional wavelets have alsobeen applied successfully to detect symmetry ofobjects such as crystals or quasicrystals.Denoising and separation of signals The waveletframe as well as the CWT give a redundantrepresentation of a data function. If, instead of theoriginal data, the redundant expression is trans-mitted, the redundancy is used to reduce the noiseincluded in the received data because the redun-dancy requires the data to belong to a subspace, andthe projection of the received data to the subspacereduces the noise component orthogonal to it. Morespecifically, the wavelet frame gives a representationof a data function as f (t) =Pj,kcj,kj,k, where theexpansion coefficients cj,k =hj,k, f (x)i satisfy thedefining equation of the subspacecj0.k0 Xcj.khj0.k0 . j.kiIf the frame coefficients are transmitted, the projec-tion operator P, which is defined on the right-handside of the above equation, reduces the noise in thereceived coefficients cj,k contaminated during thetransmission.However, this method is not applicable if thetransmitted signal is not redundant. Then somea priori criterion is necessary to discriminate betweensignal and noise. Various criteria have been pro-posed in different fields. If the signal and the noise,or plural signals have different power-law forms ofspectra, then their discrimination may be possible bythe DWT at higher-frequency region where thedifference in the magnitude of the coefficients issignificant. In this approach, the wavelets of Meyertype, that is, an orthogonal wavelet with a compactsupport in Fourier space, may be preferable becausethe wavelets of different scales are separated, at leastto some extent, in Fourier space.In fluid dynamics, the vorticity field of 2Dturbulence is found to be decomposed into coherentand incoherent vorticity fields, according as theCWT is larger than a threshold value or not,respectively. These two fields give different Fourierspectra of the velocity field (k5for coherent partwhile k3for incoherent part), showing that thecoherent structures are responsible for the deviationfrom k3predicted by the classical enstrophycascade theory. In an astronomical application, onthe other hand, the data processing is performed bya more sophisticated method taking into accountinterscale relation in the wavelet transform, becausean astronomical image contains various kindsof objects, including stars, double-stars, galaxies,nebulas, and clusters. In a medical image howevercontrast analysis is indispensable for diagnosticimaging to get a clear detailed picture of organicstructure. A scale-dependent local contrast is definedas the ratio of the CWT to that given by ananalyzing wavelet with a larger support. A multi-plicative scheme to improve the contrast is con-structed by using the local contrast.Signal CompressionSignal compression is quite an important technologyin digital communication. Speech, audio, image, anddigital video are all important fields of signal424 Wavelets: Applicationscompression, and plenty of compression methodshave been put to practical use, but we mention hereonly a few.The MRA for orthogonal wavelets gives asuccessive procedure to decompose a subspace ofL2(R) into a direct sum of two subspaces corre-sponding to higher- and lower-frequency parts; onlythe latter of which is decomposed again into itshigher- and lower-frequency parts. Algebraically,this procedure was already known before thediscovery of MRA in filter theory in electricalengineering, where a discretely sampled signal isconvoluted with a filter series to give, for example, ahigh-pass-filtered or low-pass-filtered series. Anappropriate designed pair of a high-pass and alow-pass filters followed by the downsamplingyields two new series corresponding to the higher-and lower-frequency parts, respectively, which arethen reversible by another two reconstruction filterswith the upsampling. These four filters which areoften employed in a widely used technique of sub-band coding then constitute a perfect reconstruc-tion filter bank. Under some conditions, successiveapplications of this decomposition process to theseries of lower-frequency parts, which is equivalentto the nesting structure of MRA, have been used fordata compression (quadrature mirror filter). Afamous example is a data compression system ofFBI for finger prints, consisting of wavelet codingwith scalar quantization.In MRA, however, it is only the lower-frequencyparts that are successively decomposed. If both thelower- and the higher-frequency parts are repeatedlydecomposed by the decomposition filters, then thesuccessive convolution processes correspond to adecomposition of data function by a set of wavelet-like functions, called wavelet packet, where thereare choices whether to decompose the higher- and/orthe lower-frequency parts. The best wavelet packet, inthe sense of the entropy, for example, within aspecified number of decompositions, often provideswith a powerful tool for data compression in severalareas, including speech analysis and image analysis.We also note that from the viewpoint of the best basiswhich minimizes the statistical mean square error ofthe thresholded coefficients, an orthonormal waveletbasis gives a good concentration of the energy if theoriginal signal is a piecewise smooth function super-imposed by a white noise, which is thus efficientlyremoved by thresholding the coefficients. The effi-ciency of a wavelet expansion of a signal is sometimesevaluated with the entropy of probability defined asjcj,kj2jjf jj2. A better wavelet can be selected byreducing the entropy, practically from among someset of wavelets, and its restricted expansion coefficientsgive a compressed signal. One of the systematicmethods to generate such a suitable basis is also toemploy the wavelet packets.Numerical CalculationApplication of wavelet transform, especially of theDWT, to numerical solver for a differential equation(DE) has long been studied. At the first sight, thewavelets appear to give a good DE solver becausethe wavelet expansion is generally quite efficientcompared to Fourier series due to its spatiallocalization. But its implementation to an efficientcomputer code is not so straightforward; research isstill continuing for concrete problems. Applicationof the CWT to spectral method for partial differ-ential equation (PDE) has been studied extensively.There is no wavelet which diagonalizes the differ-ential operator 00x; therefore, an efficient numer-ical method is necessary for derivatives of wavelets.Products of wavelets also yield another numericalproblem. MRA brings about mesh points which areadaptive to some extent, but finite element methodstill gives more flexible mesh points.For some scaling-invariant differential or integraloperators, including 020x2, Abel transformations,and Reisz potential, adaptive biorthogonal waveletscan be provided with block-diagonal Galerkinrepresentations, which has been applied to dataprocessing. Generally, simultaneous localization ofwavelets, both in space and in scale, leads to asparse Galerkin representation for many pseudodif-ferential operators and their inverses. A threshold-ing technique with DWT has been introduced tocoherent vortex simulation of the 2D NavierStokesequations, to reduce the relevant wavelet coeffi-cients. Another promising application of waveletoccurs as a preprocessor for an iterative Poissonsolver, where a wavelet-based preconditioning leadsto a matrix with a bounded condition number.Other Wavelets and GeneralizationsSeveral new types of wavelets have been proposed:coiflet whose scaling function has vanishingmoments giving expansion coefficients approxi-mately equal to values of the data functions, andsymlet which is an orthonormal wavelet with anearly symmetric profile. Multiwavelets are waveletswhich give a complete orthonormal system in L2space. In 2D or multidimensional applications of theDWT, separable orthonormal wavelets consisting oftensor products of 1D orthonormal wavelets arefrequently used, while nonseparable orthonormalwavelets are also available. Another generalizationWavelets: Applications 425of wavelets is the Malvar basis which is also ageneralization of local Fourier basis, and gives aperfect reconstruction. A new direction of wavelet isthe second-generation wavelets which are con-structed by lifting scheme and free from the regulardyadic procedure, and thus applicable to compactregions as S2and a finite interval.See also: Fractal Dimensions in Dynamics; ImageProcessing: Mathematics; Intermittency in Turbulence;Wavelets: Application to Turbulence; Wavelets:Mathematical Theory.Further ReadingBenedetto JJ and Frazier W (eds.) (1994) Wavelets: Mathematicsand Applications. Boca Raton, FL: CRC Press.van den Berg JC (ed.) (1999) Wavelets in Physics. Cambridge:Cambridge University Press.Daubechies I (1992) Ten Lectures on Wavelets, SIAM, CBMS61,Philadelphia.Mallat S (1998) A Wavelet Tour of Signal Processing. San Diego:Academic Press.Strang G and Nguyen T (1997) Wavelet and Filter Banks.Wellesley: Wellesley-Cambridge Press.Wavelets: Mathematical TheoryK Schneider, Universite de Provence, Marseille,FranceM Farge, Ecole Normale Supe rieure, Paris, France 2006 Elsevier Ltd. All rights reserved.IntroductionThe wavelet transform unfolds functions into time(or space) and scale, and possibly directions. Thecontinuous wavelet transform has been discoveredby Alex Grossmann and Jean Morlet who publishedthe first paper on wavelets in 1984. This mathema-tical technique, based on group theory and square-integrable representations, allows us to decompose asignal, or a field, into both space and scale, andpossibly directions. The orthogonal wavelet trans-form has been discovered by Lemarie and Meyer(1986). Then, Daubechies (1988) found orthogonalbases made of compactly supported wavelets, andMallat (1989) designed the fast wavelet transform(FWT) algorithm. Further developments were donein 1991 by Raffy Coifman, Yves Meyer, and VictorWickerhauser who introduced wavelet packets andapplied them to data compression. The developmentof wavelets has been interdisciplinary, with con-tributions coming from very different fields such asengineering (sub-band coding, quadrature mirrorfilters, timefrequency analysis), theoretical physics(coherent states of affine groups in quantummechanics), and mathematics (CalderonZygmundoperators, characterization of function spaces, har-monic analysis). Many reference textbooks areavailable, some of them we recommend are listedin the Further reading section. Meanwhile, a largespectrum of applications has grown and is stilldeveloping, ranging from signal analysis and imageprocessing via numerical analysis and turbulencemodeling to data compression.In this article, we will first define the continuouswavelet transform and then the orthogonal wavelettransform based on a multiresolution analysis.Properties of both transforms will be discussedand illustrated by examples. For a general intro-duction to wavelets, see Wavelets: Applications.Continuous Wavelet TransformLet us consider the Hilbert space of square-integr-able functions L2(R) ={f : jkf k2< 1}, equippedwith the scalar product hf , gi =RR f (x)g(x) dx(denotes the complex conjugate in the case ofcomplex-valued functions) and where the norm isdefined by kf k2 =hf , f i12.Analyzing WaveletThe starting point for the wavelet transform is tochoose a real- or complex-valued function 2L2(R), called the mother wavelet, which fulfillsthe admissibility condition,C Z 10bk

2dkjkj< 1 1wherebk Z 11x ei2kxdx 2denotes the Fourier transform, with i =1p and kthe wave number. If is integrable, that is, 2L1(R), this implies that has zero mean,Z 11x dx 0 or b0 0 3In practice, however, one also requires the wavelet to be well localized in both physical and Fourier426 Wavelets: Mathematical Theory

xm(x) dx = 0 for m = 0. M1 [4[that is, monomials up to degree M1 are exactlyreproduced. In Fourier space, this property isequivalent todmdkm(k) [k=0= 0 for m = 0. M1 [5[therefore, the Fourier transform of decayssmoothly at k =0.AnalysisFrom the mother wavelet , we generate a family ofcontinuously translated and dilated wavelets,a.b(x) = 1a x ba for a 0 and b R [6[where a denotes the dilation parameter, correspond-ing to the width of the wavelet support, and b thetranslation parameter, corresponding to the positionof the wavelet. The wavelets are normalized inenergy norm, that is, |a, b|2 =1.In Fourier space, eqn [6] readsa.b(k) =a

(ak) ei2kb[7[where the contraction with 1/a in [6] is reflected ina dilation by a [7] and the translation by b implies arotation in the complex plane.The continuous wavelet transform of a function fis then defined as the convolution of f with thewavelet family a, b:f (a. b) =

f (x)+a.b(x) dx [8[where +a, b denotes, in the case of complex-valuedwavelets, the complex conjugate.Using Parsevals identity, we getf (a. b) =

f (k)+a.b(k) dk [9[and the wavelet transform could be interpreted as afrequency decomposition using bandpass filters a, bcentered at frequencies k=ka. The wave numberk denotes the barycenter of the wavelet support inFourier spacek =

0 k[(k)[ dk

0 [(k)[ dk[10[Note that these filters have a variable width kk;therefore, when the wave number increases, thebandwidth becomes wider.SynthesisThe admissibility condition [1] implies the existenceof a finite energy reproducing kernel, which is anecessary condition for being able to reconstruct thefunction f from its wavelet coefficients ~f . One thenrecoversf (x) = 1C

0

f (a. b)a.b(x)dadba2 [11[which is the inverse wavelet transform.The wavelet transform is an isometry and one hasParsevals identity. Therefore, the wavelet transformconserves the inner product and we obtainf . g) =

f (x)g+(x) dx= 1C

0

f (a. b)g+(a. b)dadba2 [12[As a consequence, the total energy E of a signalcan be calculated either in physical space or inwavelet space, such asE =

[f (x)[2dx= 1C

0

[f (a. b)[2dadba2 [13[This formula is also the starting point for thedefinition of wavelet spectra and scalogram (seeWavelets: Application to Turbulence).ExamplesIn the following, we apply the continuous wavelettransform to different academic signals using theMorlet wavelet. The Morlet wavelet is complexvalued, and consists of a modulated Gaussian withwidth k0:(x) = (e2ixek202) e22x2k20[14[The envelope factor k0 controls the number ofoscillations in the wave packet; typically, k0 =5 isused. The correction factor ek202, to ensure itsvanishing mean, is very small and often neglected.The Fourier transform is(k) = k02 e(k202)(1k2)(ek20k1) [15[Figure 1 shows wavelet analyses of a cosine, twosines, a Dirac, and a characteristic function. BelowWavelets: Mathematical Theory 427the four signals we plot the modulus and the phaseof the corresponding wavelet coefficients.Higher DimensionsThe continuous wavelet transform can be extended tohigher dimensions in L2(Rn) in different ways. Eitherwe define spherically symmetric wavelets by setting(x) =1d([x[) for x Rnor we introduce in additionto dilations a R and translations b Rnalso rota-tions todefine wavelets witha directional sensitivity. Inthe two-dimensional case, we obtain for example,a.b.0(x) =1a R10x ba [16[where a R, b R2, and where R0 is the rotationmatrixcos 0 sin0sin 0 cos 0 [17[The analysis formula [8] then becomesf (a. b. 0) =

R2f (x)+a.b.0(x) dx [18[and for the corresponding inverse wavelet transform[11] we obtainf (x) = 1C

0

R2

20f (a. b. 0)a.b.0(x)dadbd0a3 [19[Similar constructions can be made in dimensionslarger than 2 using n1 angles of rotation.0 100 200 300 400 500 600 700 800 900 10001.510.500.511.5Cosine Two sines0 500 1000 1500 2000 2500 3000 3500 40001.510.500.511.50 100 200 300 400 500 600 700 800 900 10000.500.51Dirac0 100 200 300 400 500 600 700 800 900 10000.500.511.5Characteristic functionModulus of the wavelet coefficients Phase of the wavelet coefficients0.10.20.30.40.50.60.70.80.9200 400 600 800 1000102030405060703321012200 400 600 800 100010203040506070Modulus of the wavelet coefficients Phase of the wavelet coefficients0.10.20.30.40.50.60.70.80.9500 1000 1500 2000 2500 3000 3500 40001020304050607080903210123500 1000 1500 2000 2500 3000 3500 4000102030405060708090Modulus of the wavelet coefficients Phase of the wavelet coefficients0.050.10.150.20.250.3200 400 600 800 10001020304050607000.511.522.53200 400 600 800 100010203040506070Modulus of the wavelet coefficients0.020.040.060.080.10.120.140.160.18200 400 600 800 100010203040506070 3210123200 400 600 800 100010203040506070Phase of the wavelet coefficientsFigure 1 Examples of a one-dimensional continuous wavelet analysis using the complex-valued Morlet wavelet. Each subfigureshows on the top the function to be analyzed and below (left) the modulus of its wavelet coefficients and below (right) the phase of itswavelet coefficients.428 Wavelets: Mathematical TheoryDiscrete WaveletsFramesIt is possible to obtain a discrete set of quasiortho-gonal wavelets by sampling the scale and positionaxes a, b. For the scale a we use a logarithmicdiscretization: a is replaced by aj =aj0 , where a0 isthe sampling rate of the log a axis (a0 =( log a))and where j Z is the scale index. The position b isdiscretized linearly: b is replaced by xji =ib0aj0 ,where b0 is the sampling rate of the position axis atthe largest scale and where i Z is the positionindex. Note that the sampling rate of the positionvaries with scale, that is, for finer scales (increasing jand hence decreasing aj), the sampling rateincreases. Accordingly, we obtain the discrete wave-lets (cf. Figure 2)ji(x/) = aj12 x/xjiaj [20[and the corresponding discrete decomposition for-mula isfji = ji. f ) =

f (x/)+ji(x/) dx/ [21[Furthermore, the wavelet coefficients satisfy thefollowing estimate:A|f |22 _j.i[fji[2_ B|f |22 [22[with frame bounds B _ A 0. In the case A=B wehave a tight frame.The discrete reconstruction formula isf (x) = Cj=i=fjiji(x) R(x) [23[where C is a constant and R(x) is a residual, bothdepending on the choice of the wavelet and thesampling of the scale and position axes. For the parti-cular choice a0 =2 (which corresponds to a scalesampling by octaves) and b0 =1, we have the dyadicsampling, for which there exist special wavelets ji thatform an orthonormal basis of L2(R), that is, such thatji. j/i/ ) = cjj/ cii/ [24[where c denotes the Kronecker symbol. This meansthat the wavelets ji are orthogonal with respect totheir translates by discrete steps 2ji and their dilatesby discrete steps 2jcorresponding to octaves. Inthis case, the reconstruction formula is exact withC=1 and R=0. Note that the discrete wavelettransform has lost the invariance by translation anddilation of the continuous one.Orthogonal Wavelets and Multiresolution AnalysisThe construction of orthogonal wavelet bases and theassociated fast numerical algorithm is based on themathematical concept of multiresolution analysis(MRA). The underlying idea is to consider approx-imations fj of the function f at different scales j.The amount of information needed to go froma coarseapproximation fj to a finer resolution approximationfj1 is then described using orthogonal wavelets. Theorthogonal wavelet analysis can thus be interpreted asdecomposing the function into approximations of thefunction at coarser and coarser scales (i.e., fordecreasing j), where the differences between theapproximations are encoded using wavelets.The definition of the MRA was introduced byStephane Mallat in 1988 (Mallat 1989). Thistechnique constitutes a mathematical framework oforthogonal wavelets and the related FWT.A one-dimensional orthogonal MRA of L2(R) isdefined as a sequence of successive approximationspaces Vj, j Z, which are closed imbedded subspacesof L2(R). They verify the following conditions:Vj Vj1 \j Z [25[jZVj = L2(R) [26[jZVj = 0 [27[f (x) Vj =f (2x) Vj1 [28[ij876543210 00 100 11 3223 4 5 6 70 ......(a)(b)Figure 2 Orthogonal quintic spline wavelets j , i(x) =2j 2(2jx i ) at different scales and positions: (a) 5, 6(x),6, 32(x), 7, 108(x), and (b) corresponding wavelet coefficients.Wavelets: Mathematical Theory 429A scaling function c(x) is required to exist. Itstranslates generate a basis in each Vj, that is,VjVj = spancjiiZ [29[wherecji(x) = 2j2c(2jx i). j. i Z [30[At a given scal