The 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.

Table of Content:Chapter 1 Sobolev space and preliminaries1.1 Basic notation and function spaces1.2 Weak derivatives and Sobolev spaces1.3 Sobolev embedding theorem and interpolation formula1.4 Compactness theory1.5 Fixed point principleChapter 2 Vanishing viscosity method of nonlinear evolution system2.1 Periodic boundary and Cauchy problem for KdV system2.2 KdV system with high-order derivative term2.3 Coupled KdV systems2.4 Ferrimagnetic equations2.5 Smooth solution of Ferrimagnetic equations2.6 Coupled KdV-Schrodinger equations2.7 Singular integral and differential equations in deep water2.8 Nonlinear Schrodinger equations2.9 Nonlinear Schrodinger equations with derivative2.10 Initial value problem for Bossinesq equations2.11 Initial value problem for Langmuir turbulence equationsChapter 3 Vanishing viscosity method of quasi-linear hyperbolic system3.1 Generalized soluions to the quasi-linear hyperbolic equation3.2 Existence, uniqueness of solutions to the quasi-linear equations3.3 Convergence of solutions to the parabolic system3.4 Quasi-linear parabolic equations, viscous isentropic equations3.5 Selected results on quasi-linear parabolic equations3.6 Traveling wave soutions of some diagonal quasi-linear hyperbolic equations3.7 General solutions of diagonal quasi-linear hyperbolic equations3.8 The compensated compactness methods3.9 The existance of generalized solutions3.10 Convergence of solutions to some nonlinear dispersive equationsChapter 4 Physical viscosity and viscosity of difference scheme4.1 Indeal fluid, viscous fluid and radiation hydrodynamics equations4.2 The artificial viscosity of diffrence scheme4.2 Fundamental difference between linear and nonlinear viscosity4.4 von Neumann artificial viscosity4.5 Difference schemes with mixed viscosity4.6 Artifical viscosity problem4.7 Quanlitative analysis of singular points4.8 Numerical calcution results and analysis4.9 Local comparision of different viscosity method4.10 Implicit viscosity of PIC method4.11 2D 'artificial viscosity' problemChapter 5 Convergence of several schemes5.1 Convergence of Lax-Friedrichs difference scheme5.2 Convergence of hyperbolic equations in Lax-Friedrichs scheme5.3 Convergence of Glimm schemeChapter 6 Electric-magnethydrodynamic equations6.1 Introduction6.2 Defination of the finite energy weak solution6.3 Faedo-Galerkin approximation6.4 The vanishing viscosity limit6.5 Passing to the limit in the artifical pressure term6.6 Large-time behavior of weak solutions