Description
Book SynopsisThe book summarizes several mathematical aspects of the vanishing viscosity method and considers its applications in studying dynamical systems such as dissipative systems, hyperbolic conversion systems and nonlinear dispersion systems. Including original research results, the book demonstrates how to use such methods to solve PDEs and is an essential reference for mathematicians, physicists and engineers working in nonlinear science. Contents: Preface Sobolev Space and Preliminaries The Vanishing Viscosity Method of Some Nonlinear Evolution System The Vanishing Viscosity Method of Quasilinear Hyperbolic System Physical Viscosity and Viscosity of Difference Scheme Convergence of Lax–Friedrichs Scheme, Godunov Scheme and Glimm Scheme Electric–Magnetohydrodynamic Equations References
Table of ContentsTable of Content: Chapter 1 Sobolev space and preliminaries 1.1 Basic notation and function spaces 1.2 Weak derivatives and Sobolev spaces 1.3 Sobolev embedding theorem and interpolation formula 1.4 Compactness theory 1.5 Fixed point principle Chapter 2 Vanishing viscosity method of nonlinear evolution system 2.1 Periodic boundary and Cauchy problem for KdV system 2.2 KdV system with high-order derivative term 2.3 Coupled KdV systems 2.4 Ferrimagnetic equations 2.5 Smooth solution of Ferrimagnetic equations 2.6 Coupled KdV-Schrodinger equations 2.7 Singular integral and differential equations in deep water 2.8 Nonlinear Schrodinger equations 2.9 Nonlinear Schrodinger equations with derivative 2.10 Initial value problem for Bossinesq equations 2.11 Initial value problem for Langmuir turbulence equations Chapter 3 Vanishing viscosity method of quasi-linear hyperbolic system 3.1 Generalized soluions to the quasi-linear hyperbolic equation 3.2 Existence, uniqueness of solutions to the quasi-linear equations 3.3 Convergence of solutions to the parabolic system 3.4 Quasi-linear parabolic equations, viscous isentropic equations 3.5 Selected results on quasi-linear parabolic equations 3.6 Traveling wave soutions of some diagonal quasi-linear hyperbolic equations 3.7 General solutions of diagonal quasi-linear hyperbolic equations 3.8 The compensated compactness methods 3.9 The existance of generalized solutions 3.10 Convergence of solutions to some nonlinear dispersive equations Chapter 4 Physical viscosity and viscosity of difference scheme 4.1 Indeal fluid, viscous fluid and radiation hydrodynamics equations 4.2 The artificial viscosity of diffrence scheme 4.2 Fundamental difference between linear and nonlinear viscosity 4.4 von Neumann artificial viscosity 4.5 Difference schemes with mixed viscosity 4.6 Artifical viscosity problem 4.7 Quanlitative analysis of singular points 4.8 Numerical calcution results and analysis 4.9 Local comparision of different viscosity method 4.10 Implicit viscosity of PIC method 4.11 2D 'artificial viscosity' problem Chapter 5 Convergence of several schemes 5.1 Convergence of Lax-Friedrichs difference scheme 5.2 Convergence of hyperbolic equations in Lax-Friedrichs scheme 5.3 Convergence of Glimm scheme Chapter 6 Electric-magnethydrodynamic equations 6.1 Introduction 6.2 Defination of the finite energy weak solution 6.3 Faedo-Galerkin approximation 6.4 The vanishing viscosity limit 6.5 Passing to the limit in the artifical pressure term 6.6 Large-time behavior of weak solutions
