Chapter 1 Physical Backgrounds for Some Nonlinear Evolution
Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1
1.1 The wave equation under weak nonlinear action and KdV equation. . . . . .2
1.2 Zakharov equations and the solitons in plasma . . . . . . . . . . . . . . . . .. . . . . . . . 10
1.3 Landau-Lifshitz equations and the magnetized motion. . . . . . . . . . . . . .. . . .19
1.4 Boussinesq equation, Toda Lattice and Born-Infeld equation . . . . . . . . . .. 22
1.5 2D K-Pequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 26
Chapter 2 The Properties of the Solutions for Some Nonlinear
Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 29
2.1 The smooth solution for the initial-boundary value problem of nonlinear
Schrödin gerequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 30
2.2 The existence of the weak solution for the initial-boundary value problem
of generalized Landau-Lifshitz equations. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 34
2.2.1 The basic estimates of the linear parabolic equations . . . . . . . . . . . . . . .. . 34
2.2.2 The existence of the spin equations . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 36
2.2.3 The existence of the solution to the initial-boundary value problem of the
generalized Landau-Lifshitz equations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 39
2.3 The large time behavior for generalized KdV equation . . . . . . . . . . . . . . .. . 42
2.4 The decay estimates for the weak solution of Navier-Stokes equations . . 60
2.5 The “blowing up” phenomen on for the Cauchy problem of nonlinear
Schrödin gerequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 71
2.6 The “blow up” problem for the solutions of some semi-linear parabolic and
hyperbolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .78
2.7 The smoothness of the weak solutions for Benjamin-Ono equation . . . . . . 93
Chapter 3 Some Results for the Studies of Some Nonlinear Evolution
Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .105
3.1Nonlinear wave equations and nonlinear equations. . . . . .. . 105
3.2 KdV equation, etc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 121
3.3 Landau-Lifshitz equations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 132
Chapter 4 Similarity Solution and the Painlevé Property for Some
Nonlinear Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141
4.1Classical infinitesimal transformations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 142
4.2 Structure of Lie algebra for infinitesimal operator. . . . . . . . . . . . . .. . . . . . . 156
4.3 Non classical infinitesimal transformations. . . . . . . . . . . . . . . . . . .. . . . . . . . . . 158
4.4 A direct method for solving similarity solutions . . . . . . . . . . . . . . . .. . . . . . . 163
4.5 The Painlevé properties for some PDE. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 173
Chapter 5 Infinite Dimensional Dynamical Systems. . . . . . . . . . . . . . . . . . .182
5.1Infinite dimensional dynamical systems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 183
5.2 Some problems for infinite dimensional dynamical systems . . . . . . . . . . . . 187
5.3 Global attractor and its Hausdorff, fractal dimensions. . . . . . . . . . . . . . . .. 196
5.4 Global attractor and the bounds of Hausdorff dimensions for weak
damped KdV equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 206
5.4.1Uniform a priori estimation with respect to t. . . . . .. . . . . . . . . . . . . . . . . 207
5.5 Global attractor and the bounds of Hausdorff dimensions for weak
damped nonlinear 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 The existence of bounded absorbing set of H1 modular . . . . . . . . . . . . . .224
5.5.4 The existence of bounded absorbing set of H2 modular . . . . . . . . . . . . . .225
5.5.5 Nonlinear semi-group and long-time behavior . . . . . . . . . . . . . . . . . .. . . . . 228
5.5.6 The dimension of invariant set . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 231
5.6 Global attractor and the bounds of Hausdorff, fractal dimensions for
damped nonlinear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 238
5.6.1Linear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 238
5.6.2Nonlinear wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 243
5.6.3 The maximal attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 250
5.6.4 Dimension of the maximal attractor . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 253
5.6.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 260
5.6.6 Non-autonomous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 265
5.7 Inertial manifold for one class of nonlinear evolution equations. . . . . . . .269
5.8 Approximate inertial manifold . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 287
5.9 Nonlinear Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 296
5.10 Inertial set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 323
Chapter 6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 345
6.1 Basic notation and functional space . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 345
6.2 Sobolevembedding theorem and interpolation formula . . . . . . . . . . . . . . . . 348
6.3 Fixed point theorem . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Bibliography . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .352
Index . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 365
