Nonlinear Evolution Equations, livre ebook

The book introduces the existence, uniqueness, regularity and the long time behavior of solutions with respect to space and time, and the explosion phenomenon for some evolution equations, including the KdV equation, the nonlinear Schrödinger equation, the sine-Gordon equation, the Zakharov equations, the Landau-Lifshitz equations, the Boussinesq equation, the Navier-Stokes equations and the Newton-Boussinesq equations etc., as well as the basic concepts and research methods of infinite-dimensional dynamical systems. This book presents fundamental elements and important advances in nonlinear evolution equations. It is intended for senior university students, graduate students, postdoctoral fellows and young teachers to acquire a basic understanding of this field, while providing a reference for experienced researchers and teachers in natural sciences and engineering technology to broaden their knowledge.

Chapter 1Physical Backgrounds for Some Nonlinear Evolution

Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1

1.1 Thewave equation under weak nonlinear action and KdV equation. . . . . .2

1.2Zakharov equations and the solitons in plasma . . . . . . . . . . . . . . . . .. . . . . . . . 10

1.3Landau-Lifshitz equations and the magnetized motion. . . . . . . . . . . . . .. . . .19

1.4Boussinesq equation, Toda Lattice and Born-Infeld equation . . . . . . . . . .. 22

1.5 2D K-Pequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 26

Chapter 2The Properties of the Solutions for Some Nonlinear

EvolutionEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 29

2.1 Thesmooth solution for the initial-boundary value problem of nonlinear

Schrödingerequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 30

2.2 Theexistence of the weak solution for the initial-boundary value problem

ofgeneralized Landau-Lifshitz equations. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 34

2.2.1 Thebasic estimates of the linear parabolic equations . . . . . . . . . . . . . . .. . 34

2.2.2 Theexistence of the spin equations . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 36

2.2.3 Theexistence of the solution to the initial-boundary value problem of the

generalizedLandau-Lifshitz equations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 39

2.3 Thelarge time behavior for generalized KdV equation . . . . . . . . . . . . . . .. . 42

2.4 Thedecay estimates for the weak solution of Navier-Stokes equations . . 60

2.5 The“blowing up” phenomenon for the Cauchy problem of nonlinear

Schrödingerequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 71

2.6 The“blow up” problem for the solutions of some semi-linear parabolic and

hyperbolicequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .78

2.7 Thesmoothness of the weak solutions for Benjamin-Ono equation . . . . . . 93

Chapter 3Some Results for the Studies of Some Nonlinear Evolution

Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .105

3.1Nonlinear wave equations and nonlinear equations

. . . . . .. . 105

3.2 KdVequation, etc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 121

3.3Landau-Lifshitz equations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 132

Chapter 4Similarity Solution and the Painlevé Property for Some

NonlinearEvolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141

4.1Classical infinitesimal transformations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 142

4.2Structure of Lie algebra for infinitesimal operator. . . . . . . . . . . . . .. . . . . . . 156

4.3Nonclassical infinitesimal transformations. . . . . . . . . . . . . . . . . . .. . . . . . . . . . 158

4.4 Adirect method for solving similarity solutions . . . . . . . . . . . . . . . .. . . . . . . 163

4.5 ThePainlevé properties for some PDE. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 173

Chapter 5Infinite Dimensional Dynamical Systems. . . . . . . . . . . . . . . . . . .182

5.1Infinite dimensional dynamical systems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 183

5.2 Someproblems for infinite dimensional dynamical systems . . . . . . . . . . . . 187

5.3 Globalattractor and its Hausdorff, fractal dimensions. . . . . . . . . . . . . . . .. 196

5.4 Globalattractor and the bounds of Hausdorff dimensions for weak

damped KdVequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 206

5.4.1Uniform a priori estimation with respect to t

. . . . . .. . . . . . . . . . . . . . . . . 207

5.5 Globalattractor and the bounds of Hausdorff dimensions for weak

dampednonlinear Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 217

5.5.1Uniform a priori estimation with respect to t

. . . . . .. . . . . . . . . . . . . . . . . 218

5.5.2Transforming to Cauchy problem of the operator . . . . . . . . . . . . . . . .. . . 221

5.5.3 Theexistence of bounded absorbing set of H1 modular . . . . . . . . . . . . . .224

5.5.4 Theexistence of bounded absorbing set of H2 modular . . . . . . . . . . . . . .225

5.5.5Nonlinear semi-group and long-time behavior . . . . . . . . . . . . . . . . . .. . . . . 228

5.5.6 Thedimension of invariant set . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 231

5.6 Globalattractor and the bounds of Hausdorff, fractal dimensions for

dampednonlinear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 238

5.6.1Linear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 238

5.6.2Nonlinear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 243

5.6.3 Themaximal attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 250

5.6.4Dimension of the maximal attractor . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 253

5.6.5Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 260

5.6.6Non-autonomous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 265

5.7Inertial manifold for one class of nonlinear evolution equations. . . . . . . .269

5.8Approximate inertial manifold . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 287

5.9Nonlinear Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 296

5.10Inertial set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 323

Chapter 6Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 345

6.1 Basicnotation and functional space . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 345

6.2 Sobolevembedding theorem and interpolation formula . . . . . . . . . . . . . . . . 348

6.3 Fixed point theorem . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

Bibliography . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .352

Index . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 365