Nonlinear Evolution Equations
Published on 30. October 2023
372 pages
978-2-7598-3449-5 (ISBN)
Description
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.
More details
Edition
Digital original
Language
English
Place of publication
Les Ulis
France
Target group
Professional and scholarly
US School Grade: College Graduate Student
Edition type
Digital original
ISBN-13
978-2-7598-3449-5 (9782759834495)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Persons
Contributor: Boling GUO Guo Boling, an academician of Chinese Academy of Sciences, a renowned expert in applied mathematics and computational mathematics, research fellow of Institute of Applied Physics and Computational Mathematics, has made great achievements in qualitative and numerical study on the solutions, soliton solutions of nonlinear evolution equations, as well as infinite-dimensional dynamical systems. In addition to over 700 essays published on important journals home and abroad, he is the author of over ten books and 15 volumes of essays. He is the winner of the first prize and the third prize of National Natural Science Award, Guanghua Engineering Science and Technology Award, the Ho Leung Ho Lee (HLHL) Foundation Award, etc. --- Contributor: Fei CHEN Fei CHEN (Associate Professor, School of Mathematics and Statistics, Qingdao University) focuses on research into well-posedness and large-time behavior to solutions of some fluid mechanics equations. --- Contributor: Jing SHAO Jing SHAO (Associate Professor, Normal College, Shenyang University) carries out research on qualitative theory of fractional differential equations. --- Contributor: Ting LUO Ting LUO (Assistant professor, Master Advisor, School of Mathematical Sciences, Zhejiang Normal University) has her main research interest on stability theory of nonlinear wave equations, water waves, modeling and analysis of simplified phenomenological models, and integrable system.
Content
- Cover-1
- Cover-2
- Nonlinear Evolution Equations
- Preface
- Contents
- Chapter 1 Physical Backgrounds for Some Nonlinear Evolution Equations
- 1.1 The wave equation under weak nonlinear action and KdV equation
- 1.2 Zakharov equations and the solitons in plasma
- 1.3 Landau-Lifshitz equations and the magnetized motion
- 1.4 Boussinesq equation, Toda Lattice and Born-Infeld equation
- 1.5 2D K-P equation
- Chapter 2 The Properties of the Solutions for Some Nonlinear Evolution Equations
- 2.1 The smooth solution for the initial-boundary value problem of nonlinear Schrodinger equation
- 2.2 The existence of the weak solution for the initial-boundary value problem of generalized Landau-Lifshitz equations
- 2.2.1 The basic estimates of the linear parabolic equations
- 2.2.2 The existence of the spin equations
- 2.2.3 The existence of the solution to the initial-boundary value problem of the generalized Landau-Lifshitz equations
- 2.3 The large time behavior for generalized KdV equation
- 2.4 The decay estimates for the weak solution of Navier-Stokes equations
- 2.5 The "blowing up" phenomenon for the Cauchy problem of nonlinear Schrodinger equation
- 2.6 The "blow up" problem for the solutions of some semilinear parabolic and hyperbolic equations
- 2.7 The smoothness of the weak solutions for Benjamin-Ono equations
- Chapter 3 Some Results for the Studies of Some Nonlinear Nonlinear Evolution Equations
- 3.1 Nonlinear wave equations and nonlinear Schrodinger equations
- 3.2 KdV equation, etc.
- 3.3 Landau-Lifshitz equations
- Chapter 4 Similarity Solution and the Painleve Property for Some Nonlinear Evolution Equations
- 4.1 Classical infinitesimal transformations
- 4.2 Structure of Lie algebra for infinitesimal operator
- 4.3 Nonclassical infinitesimal transformations
- 4.4 A direct method for solving similarity solutions
- 4.5 The Painleve properties for some PDE
- Chapter 5 Infinite Dimensional Dynamical Systems
- 5.1 Infinite dimensional dynamical systems
- 5.2 Some problems for infinite dimensional dynamical systems
- 5.3 Global attractor and its Hausdorff, fractal dimensions
- 5.4 Global attractor and the bounds of Hausdorff dimensions for weak damped KdV equation
- 5.4.1 Uniform a priori estimation with respect to t
- 5.5 Global attractor and the bounds of Hausdorff dimensions for weak damped nonlinear Schrodinger equation
- 5.5.1 Uniform a priori estimation with respect to t
- 5.5.2 Transforming to Cauchy problem of the operator
- 5.5.3 The existence of bounded absorbing set of H1 modular
- 5.5.4 The existence of bounded absorbing set of H2 modular
- 5.5.5 Nonlinear semi-group and long-time behavior
- 5.5.6 The dimension of invariant set
- 5.6 Global attractor and the bounds of Hausdorff, fractal dimensions for damped nonlinear wave equation
- 5.6.1 Linear wave equation
- 5.6.2 Nonlinear wave equation
- 5.6.3 The maximal attractor
- 5.6.4 Dimension of the maximal attractor
- 5.6.5 Application
- 5.6 .6 Non-autonomous system
- 5.7 Inertial manifold for one class of nonlinear evolution equations
- 5.8 Approximate inertial manifold
- 5.9 Nonlinear Galerkin method
- 5.10 Inertial set
- Chapter 6 Appendix
- 6.1 Basic notation and functional space
- 6.2 Sobolev embedding theorem and interpolation formula
- 6.3 Fixed point theorem
- Bibliography
- Index
- Untitled
