Nonlinear Diffusion Equations

✍ Scribed by Wu Zhuoqun, Jingxue Yin, Huilai Li, Junning Zhao, Yin Jingxue, Li Huilai

• Publisher: World Scientific Pub Co Inc
• Year: 2001
• Tongue: English
• Leaves: 518
• Edition: 1
• Category: Library

✦ Synopsis

Presents the basic problems, main results, and typical methods for nonlinear diffusion equations with degeneracy. The authors, who are affiliated with Jilin University, develop Newtonian filtration equations, non-Newtonian filtration equations, general quasilinear degenerated parabolic equations of second order, and some special nonlinear diffusion equations of fourth order. Topics include properties of the free boundary, the regularity of solutions, the uniqueness of solutions, the evolution p-Laplacian equation, and the Cahn-Hilliard equation with mobility in different cases. Originally published in Chinese in 1996. No index. Annotation c. Book News, Inc., Portland, OR (booknews.com)

✦ Table of Contents

Contents
Preface
Chapter 1 Newtonian Filtration Equations
1.1 Introduction
1.1.1 Physical examples
1.1.2 Definitions of generalized solutions
1.1.3 Special solutions
1.2 Existence and Uniqueness of Solutions: One Dimensional Case
1.2.1 Uniqueness of solutions
1.2.2 Existence of solutions
1.2.3 Comparison theorems
1.2.4 Some extensions
1.3 Existence and Uniqueness of Solutions: Higher Dimensional Case
1.3.1 Comparison theorem and uniqueness of solutions
1.3.2 Existence of solutions
1.3.3 Some extensions
1.4 Regularity of Solutions: One Dimensional Case
1.4.1 Lemma
1.4.2 Regularity of solutions
1.4.3 Some extensions
1.5 Regularity of Solutions: Higher Dimensional Case
1.5.1 Generalized class B2
1.5.2 Some lemmas
1.5.3 Properties of functions in the generalized class B2
1.5.4 Holder continuity of solutions
1.6 Properties of the Free Boundary: One Dimensional Case
1.6.1 Finite propagation of disturbances
1.6.2 Localization and extinction of disturbances
1.6.3 Differential equation on the free boundary
1.6.4 Continuously differentiability of the free boundary
1.6.5 Some further results
1.7 Properties of the Free Boundary: Higher Dimensional Case
1.7.1 Monotonicity and Holder continuity of the free boundary
1.7.2 Lipschitz continuity of the free boundary
1.7.3 Differential equation on the free boundary
1.8 Initial Trace of Solutions
1.8.1 Harnack inequality
1.8.2 Main result
1.8.3 Extension of existence and uniqueness theorem
1.9 Other Problems
1.9.1 Equations with strongly nonlinear sources
1.9.2 Asymptotic properties of solutions
Chapter 2 Non-Newtonian Filtration Equations
2.1 Introduction Preliminary Knowledge
2.1.1 Introduction Physical example
2.1.2 Basic spaces and some lemmas
2.1.3 Definitions of generalized solutions
2.1.4 Special solutions
2.2 Existence of Solutions

2.2.2 The case u0 E L1loc(RN)
2.2.3 Some remarks
2.3 Harnack Inequality and the Initial Trace of Solutions
2.3.1 Local Harnack inequality
2.3.2 Global Harnack inequality
2.3.3 Initial trace of solutions
2.4 Regularity of Solutions
2.4.1 Boundedness of solutions
2.4.2 Boundedness of the gradient of solutions
2.4.3 Holder continuity of solutions
2.4.4 Holder continuity of the gradient of solutions
2.5 Uniqueness of Solutions
2.5.1 Auxiliary propositions
2.5.2 Uniqueness theorem and its proof
2.6 Properties of the Free Boundary
2.6.1 Monotonicity and Holder continuity of the free boundary
2.6.2 Lipschitz continuity of the free boundary
2.7 Other Problems
2.7.1 p-Laplacian equation with strongly nonlinear sources
2.7.2 Asymptotic properties of solutions
Chapter 3 General Quasilinear Equations of Second Order
3.1 Introduction
3.2 Weakly Degenerate Equations in One Dimension
3.2.1 Uniqueness of bounded and measurable solutions
3.2.2 Existence of continuous solutions
3.2.3 Holder continuity of solutions
3.2.4 Some extensions
3.3 Weakly Degenerate Equations in Higher Dimension
3.3.1 Existence of continuous solutions for equations with two points of degeneracy
3.3.2 Uniqueness of BV solutions
3.3.3 Existence of BV solutions
3.3.4 Some extensions
3.4 Strongly Degenerate Equations in One Dimension
3.4.1 Definitions of solutions with discontinuity
3.4.2 Interior discontinuity condition
3.4.3 Uniqueness of BV solutions of the Cauchy problem
3.4.4 Formulation of the boundary value problem
3.4.5 Boundary discontinuity condition
3.4.6 Uniqueness of BV solutions of the first boundary value problem
3.4.7 Existence of BV solutions of the first boundary value problem
3.4.8 Some extensions
3.4.9 Equations with degeneracy at infinity
3.4.10 Properties of the curves of discontinuity
3.5 Degenerate Equations in Higher Dimension without Terms of Lower Order
3.5.1 Uniqueness of bounded and integrable solutions
3.5.2 A lemma on weak convergence
3.5.3 Existence of solutions
3.5.4 Finite propagation of disturbances
3.6 General Strongly Degenerate Equations in Higher Dimension
3.6.1 Existence of BV solutions
3.6.2 Some extensions
3.7 Appendix Classes BV and BVx
Chapter 4 Nonlinear Diffusion Equations of Higher Order
4.1 Introduction
4.2 Similarity Solutions of a Fourth Order Equation
4.2.1 Definition of similarity solutions
4.2.2 Existence and uniqueness of global solutions of the Cauchy problem
4.2.3 Regularity of solutions
4.2.4 Properties of solutions at zero points
4.2.5 Properties of unbounded solutions
4.2.6 Bounded solutions on the half line
4.2.7 Bounded solutions on the whole line
4.2.8 Properties of solutions in typical cases k = 1,2,3,4
4.2.9 Behavior of similarity solutions as t -> 0+
4.3 Equations with Double-Degeneracy
4.3.1 Existence of solutions
4.3.2 Uniqueness of solutions
4.3.3 Weighted energy equality of solutions
4.3.4 Some auxiliary inequalities
4.3.5 Finite propagation of disturbances
4.3.6 Asymptotic behavior of solutions
4.3.7 Extinction of solutions at finite time
4.3.8 Nonexistence of nonnegative solutions
4.3.9 Infinite propagation case
4.4 Cahn-Hilliard Equation with Constant Mobility
4.4.1 Existence of classical solutions
4.4.2 Blowing-up of solutions
4.4.3 Global existence of solutions for small initial value
4.5 Cahn-Hilliard Equations with Positive Concentration Dependent Mobility
4.5.1 A modified Campanato space
4.5.2 Holder norm estimates for a linear problem
4.5.3 Zero potential case
4.5.4 General case
4.6 Thin Film Equation
4.6.1 Definition of generalized solutions
4.6.2 Approximate solutions
4.6.3 Existence of solutions
4.6.4 Nonnegativity of solutions
4.6.5 Zeros of nonnegative solutions
4.6.6 Regularity of solutions
4.6.7 Monotonicity of the support of solutions
4.7 Cahn-Hilliard Equation with Degenerate Mobility
4.7.1 Models with degenerate mobility
4.7.2 Definition of physical solutions
4.7.3 Existence of solutions
4.7.4 Physical solutions
Bibliography