Vanishing Viscosity Method : Solutions to Nonlinear Systems.

Intro -- Contents -- 1 Sobolev Space and Preliminaries -- 1.1 Basic Notation and Function Spaces -- 1.1.1 Basic Notation -- 1.1.2 Function Spaces -- 1.1.3 Some Basic Inequalities -- 1.2 Weak Derivatives and Its Properties, Wm p (K) and Hj,p(K) Spaces -- 1.3 Sobolev Embedding Theorem and Interpolation Formula -- 1.4 Compactness Theory -- 1.5 Fixed Point Principle -- 2 The Vanishing Viscosity Method of Some Nonlinear Evolution System -- 2.1 Periodic Boundary and Cauchy Problem for High-Order Generalized KdV System in Dimension One -- 2.2 Some KdV System with High-Order Derivative Term -- 2.3 High-Order Multivariable KdV Systems and Hirota Coupled KdV Systems -- 2.4 Initial Boundary Value Problem for Ferrimagnetic Equations -- 2.5 Existence and Uniqueness of Smooth Solution to Ferrimagnetic Equations and Other Problems for High-Dimensional Ferrimagnetic Equations with Nonlinear Boundary Conditions -- 2.6 Periodic Boundary Problem and Initial Value Problem for the Coupling KdV-Schrödinger Equations -- 2.7 Initial Value Problem for the Nonlinear Singular Integral and Differential Equations in Deep Water -- 2.8 Initial Value Problem for the Nonlinear Schrödinger Equations -- 2.9 Initial Value Problem and Boundary Value Problem for the Nonlinear Schrödinger Equation with Derivative -- 2.10 Initial Value Problem for Boussinesq Equations -- 2.11 Initial Value Problem for Langmuir Turbulence Equations -- 3 The Vanishing Viscosity Method of Quasilinear Hyperbolic System -- 3.1 Generalized Solutions to the First-Order Quasilinear Hyperbolic Equation in One Dimension -- 3.2 Existence and Uniqueness of the General Solution to First-Order Multivariable Quasilinear Equations -- 3.3 Convergence of Solutions to the Multidimensional Linear Parabolic System with Small Parameter.

3.4 On Gradient Quasilinear Parabolic Equations and Viscous Isentropic Gas Hydrodynamic Equations -- 3.5 On Some Results of Some Quasilinear Parabolic Equations -- 3.6 On Traveling Wave Solutions of Some Quasilinear Parabolic Equations with Small Parameter -- 3.7 On General Solutions of Some Diagonal Quasilinear Hyperbolic Equations -- 3.8 The Compensated Compactness Method -- 3.9 The Existence of Generalized Solutions for the First-Order Quasilinear Hyperbolic System -- 3.10 Convergence of Solutions to Some Nonlinear Dispersive Equations -- 4 Physical Viscosity and Viscosity of Difference Scheme -- 4.1 Ideal Fluid, Viscous Fluid and Radiation Hydrodynamic Equations -- 4.2 The Artificial Viscosity of Difference Scheme -- 4.3 The Fundamental Difference Between Linear and Nonlinear Viscosity Qualitatively -- 4.4 von Neumann Artificial Viscosity and Godunov Scheme Implicit Viscosity -- 4.5 Several Difference Schemes with Mixed Viscosity -- 4.6 Artificial Viscosity Problem for Hydrodynamic Equations with Radiation Conductivity Transfer Term -- 4.7 Qualitative Analysis of Singular Points of Some Artificial Viscosity Equation -- 4.8 Numerical Calculation Results and Analysis -- 4.9 Local Comparison of Different Viscosity Method to Initial Discontinuity Problem for One-Dimensional Radiation Hydrodynamic Equations with Different Medium -- 4.10 Implicit Viscosity of PIC Method -- 4.11 Two-Dimensional "Artificial Viscosity" Problem -- 5 Convergence of Lax-Friedrichs Scheme, Godunov Scheme and Glimm Scheme -- 5.1 Convergence of Lax-Friedrichs Difference Scheme -- 5.2 The Convergence of Hyperbolic Equations in Lax-Friedrichs Scheme -- 5.3 Convergence of Glimm Scheme -- 6 Electric-Magnetohydrodynamic Equations -- 6.1 Introduction -- 6.2 Definition of the Finite Energy Weak Solution -- 6.3 The Faedo-Galerkin Approximation -- 6.4 The Vanishing Viscosity Limit.

6.5 Passing to the Limit in the Artificial Pressure Term -- 6.5.1 Passing to the Limit -- 6.5.2 The Effective Viscous Flux -- 6.5.3 The Amplitude of Oscillations -- 6.5.4 The Renormalized Solutions -- 6.5.5 Strong Convergence of the Density -- 6.6 Large-Time Behavior of Weak Solutions -- References.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.