Contents Preface Chapter 1 Background of Ginzburg-Landau Equations 1 1.1 The Benard convection 1 1.2 The Couette-Taylor flow 6 1.3 The plane Poiseuille flow 12 1.4 The turbulent problem in chemical reaction 15 1.5 Transition from KS equation to Ginzburg-Landau equation 21 1.6 Ginzburg-Landau models in superconductivity 22 Chapter 2 Global Solutions and Global Attractors for One Dimensional Ginzburg-Landau Equations 27 2.1 Global solutions and global attractors 27 2.2 Analysis for traveling wave solutions 39 2.3 Instability of the quasi-periodic solutions 53 2.4 Nonlinear stability of the plane waves 64 2.5 Finite dimensional inertial manifolds 72 2.6 Exponential attractors 89 2.7 Structure of the inertial manifolds 94 2.8 Gevrey regularity 117 2.9 Determining nodes 129 2.10 Dynamical system structure and numerical analysis 138 2.11 Slow periodic solutions 146 2.12 Stability of traveling wave solutions 164 2.13 Upper bound estimates of winding numbers 176 2.14 Discrete attractors and their dimension estimates 186 2.15 Stability criterion for perturbed cubic-quintic nonlinear Schrodinger equations 206 2.16 Nonlinear instability of the plane waves 229 Chapter 3 Global Solutions and Asymptotic Behavior for Higher Dimensional Ginzburg-Landau Equations 237 3.1 Global solutions 237 3.2 Cauchy problem in local spaces 275 3.3 Global attractors for 2D case 307 3.4 The dynamical length 313 3.5 Hausdor measures of level sets of solutions 327 3.6 Global attractors for 2D derivative Ginzburg-Landau equation 341 3.7 Gevrey regularity and approximate inertial manifolds 356 3.8 Global attractors for the case of unbounded domain 368 3.9 Time periodic solutions 383 3.10 Limits to nonlinear Schr.odinger equations 392 3.11 Existence of almost periodic solutions 407 Chapter 4 Ginzburg-Landau Equations in Superconductivity 422 4.1 Cauchy problem 422 4.2 Global attractors 433 4.3 Hyperbolic Ginzburg-Landau Equations 440 4.4 Instability of symmetric vortices 446 Chapter 5 Ginzburg-Landau Model Equations 473 5.1 The case of deg(g,*Ω)=0 474 5.2 The case of deg(g,*Ω)≠0 500 5.3 Equations of Ginzburg-Landau heat flows 543 5.4 Ginzburg-Landau equations and mean curvature flows 559 References 588
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