فهرست مطالب
Contents\n1 Sobolev Space and Preliminaries\n 1.1 Basic Notation and Function Spaces\n 1.1.1 Basic Notation\n 1.1.2 Function Spaces\n 1.1.3 Some Basic Inequalities\n 1.2 Weak Derivatives and Its Properties, Wm p (K) and Hj,p(K) Spaces\n 1.3 Sobolev Embedding Theorem and Interpolation Formula\n 1.4 Compactness Theory\n 1.5 Fixed Point Principle\n2 The Vanishing Viscosity Method of Some Nonlinear Evolution System\n 2.1 Periodic Boundary and Cauchy Problem for High-Order Generalized KdV System in Dimension One\n 2.2 Some KdV System with High-Order Derivative Term\n 2.3 High-Order Multivariable KdV Systems and Hirota Coupled KdV Systems\n 2.4 Initial Boundary Value Problem for Ferrimagnetic Equations\n 2.5 Existence and Uniqueness of Smooth Solution to Ferrimagnetic Equations and Other Problems for High-Dimensional Ferrimagnetic Equations with Nonlinear Boundary Conditions\n 2.6 Periodic Boundary Problem and Initial Value Problem for the Coupling KdV–Schrödinger Equations\n 2.7 Initial Value Problem for the Nonlinear Singular Integral and Differential Equations in Deep Water\n 2.8 Initial Value Problem for the Nonlinear Schrödinger Equations\n 2.9 Initial Value Problem and Boundary Value Problem for the Nonlinear Schrödinger Equation with Derivative\n 2.10 Initial Value Problem for Boussinesq Equations\n 2.11 Initial Value Problem for Langmuir Turbulence Equations\n3 The Vanishing Viscosity Method of Quasilinear Hyperbolic System\n 3.1 Generalized Solutions to the First-Order Quasilinear Hyperbolic Equation in One Dimension\n 3.2 Existence and Uniqueness of the General Solution to First-Order Multivariable Quasilinear Equations\n 3.3 Convergence of Solutions to the Multidimensional Linear Parabolic System with Small Parameter\n 3.4 On Gradient Quasilinear Parabolic Equations and Viscous Isentropic Gas Hydrodynamic Equations\n 3.5 On Some Results of Some Quasilinear Parabolic Equations\n 3.6 On Traveling Wave Solutions of Some Quasilinear Parabolic Equations with Small Parameter\n 3.7 On General Solutions of Some Diagonal Quasilinear Hyperbolic Equations\n 3.8 The Compensated Compactness Method\n 3.9 The Existence of Generalized Solutions for the First-Order Quasilinear Hyperbolic System\n 3.10 Convergence of Solutions to Some Nonlinear Dispersive Equations\n4 Physical Viscosity and Viscosity of Difference Scheme\n 4.1 Ideal Fluid, Viscous Fluid and Radiation Hydrodynamic Equations\n 4.2 The Artificial Viscosity of Difference Scheme\n 4.3 The Fundamental Difference Between Linear and Nonlinear Viscosity Qualitatively\n 4.4 von Neumann Artificial Viscosity and Godunov Scheme Implicit Viscosity\n 4.5 Several Difference Schemes with Mixed Viscosity\n 4.6 Artificial Viscosity Problem for Hydrodynamic Equations with Radiation Conductivity Transfer Term\n 4.7 Qualitative Analysis of Singular Points of Some Artificial Viscosity Equation\n 4.8 Numerical Calculation Results and Analysis\n 4.9 Local Comparison of Different Viscosity Method to Initial Discontinuity Problem for One-Dimensional Radiation Hydrodynamic Equations with Different Medium\n 4.10 Implicit Viscosity of PIC Method\n 4.11 Two-Dimensional “Artificial Viscosity” Problem\n5 Convergence of Lax–Friedrichs Scheme, Godunov Scheme and Glimm Scheme\n 5.1 Convergence of Lax–Friedrichs Difference Scheme\n 5.2 The Convergence of Hyperbolic Equations in Lax–Friedrichs Scheme\n 5.3 Convergence of Glimm Scheme\n6 Electric–Magnetohydrodynamic Equations\n 6.1 Introduction\n 6.2 Definition of the Finite Energy Weak Solution\n 6.3 The Faedo–Galerkin Approximation\n 6.4 The Vanishing Viscosity Limit\n 6.5 Passing to the Limit in the Artificial Pressure Term\n 6.5.1 Passing to the Limit\n 6.5.2 The Effective Viscous Flux\n 6.5.3 The Amplitude of Oscillations\n 6.5.4 The Renormalized Solutions\n 6.5.5 Strong Convergence of the Density\n 6.6 Large-Time Behavior of Weak Solutions\nReferences
