Invariant Manifold Theory for Hydrodynamic Transition - Softcover

Sritharan, S.S.

Softcover

ISBN 10: 048682828X ISBN 13: 9780486828282

Verlag: Dover Publications Inc., 2019

Inhaltsangabe

"Invariant manifold theory serves as a link between dynamical systems theory and turbulence phenomena. This volume consists of research notes by author S. S. Sritharan that develop a theory for the Navier-Stokes equations in bounded and certain unbounded geometries. The main results include spectral theorems and analyticity theorems for semigroups and invariant manifolds.""This monograph contains a lot of useful information, including much that cannot be found in the standard texts on the Navier-Stokes equations,"" observed MathSciNet, adding ""the book is well worth the reader's attention."" The treatment is suitable for researchers and graduate students in the areas of chaos and turbulence theory, hydrodynamic stability, dynamical systems, partial differential equations, and control theory. Topics include the governing equations and the functional framework, the linearized operator and its spectral properties, the monodromy operator and its properties, the nonlinear hydrodynamic semigroup, invariant cone theorem, and invariant manifold theorem. Two helpful appendixes conclude the text."

Über diesen Titel

Über die Autorin bzw. den Autor

Sivaguru S. Sritharan is Provost and Academic Dean at the Air Force Institute of Technology, located at Ohio's Wright-Patterson Air Force Base. He also serves as the Air Force Institute of Technology's Chief Operating Officer and Chief Academic Officer. He received his Ph.D. in Applied Mathematics from the University of Arizona and was the editor of Optimal Control of Viscous Flow .

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Invariant Manifold Theory for Hydrodynamic Transition

By S. S. Sritharan

Dover Publications, Inc.

Copyright © 2019 S. S. Sritharan
All rights reserved.

ISBN: 978-0-486-82828-2

Contents

Preface, vii, 
1 Introduction, 1, 
2 The Governing Equations and the Functional Framework, 5, 
3 The Linearized Operator and its Spectral Properties, 21, 
4 The Monodromy Operator and its Properties, 39, 
5 The Nonlinear Hydrodynamic Semigroup, 71, 
6 Invariant Cone Theorem, 85, 
7 Invariant Manifold Theorem, 93, 
A Global Attractors for Two-Dimensional Viscous Flows, 111, 
B Implications of Group Action, 121, 
Bibliography, 131, 
Index, 139, 

CHAPTER 1

Introduction

An invariant manifold theory for the Navier-Stokes equations would lay a bridge between the theory of finite dimensional dynamical systems and the onset dynamics of turbulence. In this work we will develop such a theory for hydrodynamic motions in bounded containers. Although the main focus of this work is on bounded domains we will discuss extensions to unbounded domains in each section and point out open problems. In simple words, invariant manifold theory identifies the active and slave modes of a particular solution orbit with respect to a basic solution orbit.

It has been proven by Ladyzhenkaya that the longtime behavior of viscous flows in two-dimensional bounded domains can be characterized by a compact attractor. This attractor contains (at least) all the stationary, time periodic and quasiperiodic solutions. Moreover, the Navier-Stokes equations define a dynamical system in this attractor. This means that the solution (in this attractor) is smooth and defined for positive as well as negative times. Numerical approximations and mathematical regularizations of the Navier-Stokes equations produce upper semi-continuous global attractors which converge to the global attractor of the conventional system as the approximation (or the regularization) parameter approaches zero. These nice properties of the global attractors make them the central object of research for turbulence theory. It is also of interest to study the orbits nearby such attractors.

For the case of three-dimensional flows, existence of a compact global attractor has not been proven yet. As one might expect, this problem is connected with the global unique solvability theorem which is not available for the three-dimensional case. Hence, the fundamental step in the dynamical systems theory for the Navier-Stokes equations is the global unique solvability theorem which is, in some sense, the same as the continuous dependence theorem. Hadamard's notion of well posedness requires the existence, uniqueness and continuous dependence of the solution with respect to data. Unfortunately, a complete answer to this basic question is available only for two-dimensional (time dependent) viscous flow in bounded domains and for certain unbounded flow problems. For these problems it is possible to prove that the solution exists, unique, is holomorphic in time in the neighborhood of the positive real axis and is Fréchet analytic in initial and boundary data. The solvability problem and hence the dynamical systems theoretical description is not complete for the other cases. The global unique solvability problem for viscous flow in three-dimensional bounded (or unbounded) domains is now regarded as one of the profound open problems in mathematical science. In the simplest setting this problem can be described as follows. Consider the viscous flow in a three-dimensional bounded container with homogeneous (nonslip) boundary conditions. Motion is initiated by prescribing an initial velocity distribution with a finite but arbitrary amount of energy. In other words the initial velocity has finite L2(Ω) norm. For this case there is a famous theorem due to Hopf which provides the existence of a weak solution with the following properties. The energy E(t) or the L2(Ω) norm of the velocity field is bounded for all subsequent times (E(t ≤ E(0)) and the enstrophy EN (t) which is the square integral of the vorticity (also the Dirichlet integral of the velocity) is integrable in time. This defines what is now called the Leray-Hopf-Ladyzhenskaya topology (L∞(0, T; L2(Ω)) [intersection] L2 (0, T; H10 (Ω))) which has become a standard mathematical framework in the treatment of parabolic partial differential equations. Unfortunately, this information is not sufficient to prove the uniqueness of this class of weak solutions. In two dimensions, however, this is in fact sufficient as demonstrated by Lions and Prodi [58]. For the two-dimensional case of this problem, we can show that the solution is holomorphic in the neighborhood of the positive time axis and Fréchet analytic in the initial data. For the three-dimensional case, it is possible to establish the uniqueness of the Hopf class solutions if it is known that the enstrophy EN (t) is square integrable in time. In other words the L2(0, T) norm of EN (t) is bounded (meaning we need a uniform bound on solution in the norm L4(0, T; H10 (Ω)).). Is this possible? Since this additional requirement is expressed in terms of the vorticity in the flow, it is certainly reasonable to compare the nature of vorticity dynamics in two and three dimensions. In three-dimensional flows there are additional features such as vortex stretching and knottedness of vortex filaments (which is associated with nontrivial helicity [68]). Hence, the resolution of the global unique solvability theorem may require a detailed understanding of the solutions of the three-dimensional Euler equations.

Let us now formulate a slightly different problem. We consider again the three-dimensional viscous flow problem with homogeneous boundary data and an initial velocity with finite energy and enstrophy. This time we will introduce a forcing (perhaps a distributed forcing in the sense of control theory) at the right-hand side of the momentum equations. Here again, if the force is square integrable in space and time then we have a Hopf class weak solution with the same properties as above. In this case, however, Fursikov [23] has shown that there is a set of such forces (included in the class forces that are divergence free and square integrable in space and time) for which the additional requirement EN (t) [member of] L2(0, T) is satisfied. For this case the problem is uniquely solvable. It is however not known whether this class of forces includes zero. The Fursikov theorem, however, is a generic solvability theorem. Thus in order to resolve the three-dimensional homogeneous Navier-Stokes problem we only need to introduce a force which is arbitrarily close to zero in a suitable topology.

In light of Fursikov's result we may say that a possible method of resolving the global unique solvability problem is to show that his class of forces includes zero as an element. Mathematical regularizations seem to provide another promising method. Here we add (to the momentum equations) higher order terms such as Laplacian square with artificial viscosity coefficients. For such a system, the nonlinearity (the inertia term) exhibits much better behavior and allows us to establish global unique solvability up to several dimensions. The task then would be to show that, as the regularization parameter approaches zero, the solution of the regularized system converges in some topology to the solution of the conventional system. This has been accomplished for the case of low Reynolds numbers for three-dimensional problems and for arbitrary Reynolds numbers for two-dimensional problems.

In this monograph, however, we will develop a dynamical systems theory about a smooth basic solution and hence global-in-time solvability theorems with small data are sufficient for most purposes. We will consider stationary and time periodic smooth solutions and study the nearby solutions. The Reynolds number is held fixed and hence does not play an explicit role.

In the second chapter we will present the functional framework used in this monograph. Analysis of the Stokes problem with various boundary conditions is given. We define an accretive self-adjoint operator called the Stokes operator and characterize its fractional powers. The central result used here is the Cattabriga regularity theorem. A functional framework for unbounded domains is discussed along with a general theory of the Stokes operator.

In the third chapter we study the linearizations (of the stationary Navier-Stokes equations) about a stationary basic solution. We will also provide the existence and regularity results for stationary basic solutions. We then consider the linearized operator about a smooth basic field and establish its spectral properties. Although the spectral theorem 3.6 is stated for the case of stationary basic flow in fact it applies (for each time t) if the basic flow is time dependent. The completeness theorem 3.6 for the eigenfunctions is needed in any constructive procedure that uses the invariant manifold method to compute the bifurcating solutions. Mathematical theory of exterior hydrodynamics including spectral theory is also discussed.

In chapter four we study the linear evolution problem obtained by linearizing the Navier-Stokes equations about a smooth time dependent basic flow. The main results in this section are the regularity and spectral properties of the monodromy operator. We also provide existence theorems for time periodic solutions.

The nonlinear semigroup associated with the Navier-Stokes equations is considered in chapter five. We establish the analyticity of the solution map with respect to the initial data. This property of the solution map enables us to establish the analyticity of the invariant manifolds in chapter seven. For three-dimensional flows, because of the lack of global unique solvability theorem for large data, the semigroup can be defined only up to a finite maximal time determined by the size of the initial data. However, it is possible to take a small enough initial data (in certain norm) so that the maximal time is infinity. For two-dimensional flows, of course, the maximal time is infinity for arbitrary initial data.

In chapter six we establish the principle of linearized stability for stationary and time periodic basic solutions. Our result is independent of a specific topology. The Lyapunov instability is characterized using the invariant cone concept introduced by Kirchgassner and Scheurle.

In chapter seven we refine this result and establish the existence and uniqueness of invariant manifolds. The central result of this chapter is the analyticity of these manifolds. We prove this result by working with classes of analytic maps throughout this section. The analyticity theorem provides us with an analytic coordinate chart for the unstable manifold and a convergent expansion method to compute bifurcating solutions.

In Appendix A, we present Ladyzhenskaya's theory of global attractors for two-dimensional viscous flows in bounded domains.

In Appendix B, we will derive certain implications of group action on the invariant manifold theory.

Future directions of this subject will expected to bring together ideas from ergodic theory, control theory and large deviations to the dynamical systems perspective given in this book. Extending the theory to compressible viscous flow appears to be promising based on the recent literature on solvability theory of global solutions for small data and also on time periodic solutions.

CHAPTER 2

The Governing Equations and the Functional Framework

We will first formulate the mathematical problem of time dependent viscous flow in bounded containers with prescribed boundary velocity distributions. Formulations for other problems would require additional conditions such as far-field and flux conditions and they will be discussed later in this chapter. Let Ω [subset] Rn, n = 2 or 3 be an open bounded set of class Cr, r ≥ 2. The smoothness requirement here is not crucial to the developments of this monograph. We will consider the motion of a viscous incompressible fluid in Ω with the prescribed initial velocity field u0 and a time dependent boundary distribution ub. Let u, p and v, respectively, denote the velocity field, pressure field and kinematic viscosity. The problem is to find (u, p): Ω × (0, ∞) [right arrow] Rn × R such that

[MATHEMATICAL EXPRESSION OMITTED] (2.1)

Here dΣ is the surface area vector. Note that if Ω is multi-connected then the divergence free condition would dictate that the sum of the flux through all the components of [partial derivative]Ω be zero. However, most of the existence theorems in the literature (for steady as well as unsteady flows) would require that the flux through each individual component of [partial derivatie]Ω be separately zero.

Let (U(x, t), P(x, t)) be the basic solution field satisfying the governing equations and the boundary conditions. We are interested in studying the solution orbits nearby this given orbit in an appropriate function space. A case of particular interest is when the basic solution orbit belongs to the global attractor (see appendixA for a discussion on global attractors for the two-dimensional case). Let us introduce the change of variables u = U + v and p = P +q in (2.1) to get

[MATHEMATICAL EXPRESSION OMITTED] (2.2)

and

v(x, 0) = v0(x) for x [member of] Ω.

We will introduce the following function spaces:

[MATHEMATICAL EXPRESSION OMITTED]

Here Hm(Ω) is the Sobolev space of vector fields whose distributional derivatives of order up to m are in L2(Ω). It can be shown that for bounded domains the spaces H and V are respectively the closures of j(Ω) in the norms of L2(Ω) and H1(Ω). One should note that due to the Poincaré's lemma (which applies when Ω is bounded in one direction) the norm of H1(Ω) and the norm obtained from the Dirichlet integral [parallel]u[parallel]2 = ∫Ω [nabla]u · [nabla]u dx are equivalent in V. The trace u(x) · n, x [member of] [partial derivative]Ω can be characterized as an element of H-1/2 ([partial derivative]Ω) when u and div u are square integrable.

Let us now define the Stokes operator. Consider the symmetric bilinear form a(·, ·) defined by

a(u, v) = v ∫Ω [nabla]u · [nabla]v dx.

Then it is immediate that

[MATHEMATICAL EXPRESSION OMITTED]

by Schwartz inequality and

[MATHEMATICAL EXPRESSION OMITTED]

Hence by Lax-Milgram lemma there exists an isomorphism A [member of] L(V; V') such that

[MATHEMATICAL EXPRESSION OMITTED]

where V' = L(V; R) the dual of V.

We can define a restriction of this operator [??] in the following way. For a given u [member of] V, if there exists g [member of] H such that

a(u, v) = (g, v)H, [for all]v [member of] V

then [??]u = g and u = D([??]). The operator [MATHEMATICAL EXPRESSION OMITTED]); H) is again an isomorphism. The explicit form of D([??]) can be obtained from the Cattabriga regularity theorem:

Theorem 2.1.Let Ω [subset] Rn, n = 2, 3 be bounded and of class Cr, r = max(s + 2, 2) and let g [member of] Hs(Ω), s ≥ -1 be a given vector field. Then for the Stokes problem

[MATHEMATICAL EXPRESSION OMITTED]

the unique solution (u, q) [member of] [Hs+2(Ω) [intersection] V] × Hs+1(Ω)/R and satisfies the estimate

[MATHEMATICAL EXPRESSION OMITTED]

Hence we write D([??]) = H2(Ω) [intersection] V. The operator A [mmeber of] L(V; V') [intersection] L(D(A); H) is m-accretive and self-adjoint.

Let us use the Riesz representation theorem to identify H with its dual H'. This gives us the continuous and dense embeddings:

D(A) [subset] V [subset] H [equivalent to] H' [subset] V' [subset] D(A)'.

Rellich's lemma leads to the compactness of these embeddings.

(Continues...)

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Details zur Ausgabe

• Verlag: Dover Publications Inc.
• Erscheinungsdatum: 2019
• ISBN 10: 048682828X
• ISBN 13: 9780486828282
• Einband: Tapa blanda
• Sprache: Englisch
• Anzahl der Seiten: 176
• Kontakt zum Hersteller: Nicht verfügbar