Nonlinear Partial Differential Equations

, by Giga, Mi-Ho; Giga, Yoshikazu; Saal, Jurgen

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ISBN: 9780817641733 | 0817641734
Cover: Hardcover
Copyright: 9/30/2010

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The main focus of this textbook, in two parts, is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. The exposition moves systematically from the basic to more sophisticated concepts with recent developments and several open problems. With challenging exercises, examples, and illustrations to help explain the rigorous analytic basis for the Navier-Stokes equations, mean curvature flow equations, and other important equations describing real phenomena, this book is written for graduate students and researchers, not only in mathematics but also in other disciplines.

Preface	p. xiii
Asymptotic Behavior of Solutions of Partial Differential Equations
Behavior near Time Infinity of Solutions of the Heat Equation	p. 3
Asymptotic Behavior of Solutions near Time Infinity	p. 3
Decay Estimate of Solutions	p. 6
L^p-L^q Estimates	p. 8
Derivative L^p-L^q Estimates	p. 8
Theorem on Asymptotic Behavior Near Time Infinity	p. 10
Proof Using Representation Formula of Solutions	p. 11
Integral Form of the Mean Value Theorem	p. 12
Structure of Equations and Self-Similar Solutions	p. 13
Invariance Under Scaling	p. 13
Conserved Quantity for the Heat Equation	p. 14
Scaling Transformation Preserving the Conserved Quantity	p. 15
Summary of Properties of a Scaling Transformation	p. 15
Self-Similar Solutions	p. 16
Expression of Asymptotic Formula Using Scaling Transformations	p. 16
Idea of the Proof Based on Scaling Transformation	p. 17
Compactness	p. 18
Family of Functions Consisting of Continuous Functions	p. 19
Ascoli-Arzelà-type Compactness Theorem	p. 22
Relative Compactness of a Family of Scaled Functions	p. 22
Decay Estimates in Space Variables	p. 25
Existence of Convergent Subsequences	p. 26
Lemma	p. 27
Characterization of Limit Functions	p. 27
Limit of the Initial Data	p. 28
Weak Form of the Initial Value Problem for the Heat Equation	p. 29
Weak Solutions for the Initial Value Problem	p. 30
Limit of a Sequence of Solutions to the Heat Equation	p. 31
Characterization of the Limit of a Family of Scaled Functions	p. 32
Uniqueness Theorem When Initial Data is the Delta Function	p. 33
Completion of the Proof of Asymptotic Formula (1 9) Based on Scaling Transformation	p. 34
Remark on Uniqueness Theorem	p. 34
Behavior Near Time Infinity of Solutions of the Vorticity Equations	p. 37
Navier-Stokes Equations and Vorticity Equations	p. 38
Vorticity	p. 39
Vorticity and Velocity	p. 40
Biot-Savart Law	p. 41
Derivation of the Vorticity Equations	p. 42
Asymptotic Behavior Near Time Infinity	p. 42
Unique Existence Theorem	p. 43
Theorem for Asymptotic Behavior of the Vorticity	p. 44
Scaling Invariance	p. 44
Conservation of the Total Circulation	p. 45
Rotationally Symmetric Self-Similar Solutions	p. 46
Global L^q-L¹ Estimates for Solutions of the Heat Equation with a Transport Term	p. 47
Fundamental L^q-L^r Estimates	p. 47
Change Ratio of L^r-Norm per Time: Integral Identities	p. 48
Nonincrease of L¹-Norm	p. 49
Application of the Nash Inequality	p. 50
Proof of Fundamental L^q-L¹ Estimates	p. 53
Extension of Fundamental L^q-L¹ Estimates	p. 55
Maximum Principle	p. 55
Preservation of Nonnegativity	p. 56
Estimates for Solutions of Vorticity Equations	p. 58
Estimates for Vorticity and Velocity	p. 58
Estimates for Derivatives of the Vorticity	p. 62
Decay Estimates for the Vorticity in Spatial Variables	p. 68
Proof of the Asymptotic Formula	p. 72
Characterization of the Limit Function as a Weak Solution	p. 73
Estimates for the Limit Function	p. 76
Integral Equation Satisfied by Weak Solutions	p. 80
Uniqueness of Solutions of Limit Equations	p. 81
Completion of the Proof of the Asymptotic Formula	p. 83
Formation of the Burgers Vortex	p. 84
Convergence to the Burgers Vortex	p. 85
Asymmetric Burgers Vortices	p. 87
Self-Similar Solutions of the Navier-Stokes Equations and Related Topics	p. 88
Short History of Research on Asymptotic Behavior of Vorticity	p. 89
Problems of Existence of Solutions	p. 91
Self-Similar Solutions	p. 93
Uniqueness of the Limit Equation for Large Circulation	p. 97
Uniqueness of Weak Solutions	p. 97
Relative Entropy	p. 98
Boundedness of the Entropy	p. 100
Rescaling	p. 100
Proof of the Uniqueness Theorem	p. 101
Remark on Asymptotic Behavior of the Vorticity	p. 102
Self-Similar Solutions for Various Equations	p. 105
Porous Medium Equation	p. 105
Self-Similar Solutions Preserving Total Mass	p. 107
Weak Solutions	p. 108
Asymptotic Formula	p. 109
Roles of Backward Self-Similar Solutions	p. 109
Axisymmetric Mean Curvature Flow Equation	p. 110
Backward Self-Similar Solutions and Similarity Variables	p. 111
Nonexistence of Nontrivial Self-Similar Solutions	p. 114
Asymptotic Behavior of Solutions Near Pinching Points	p. 116
Monotonicity Formula	p. 121
The Cases of a Semilinear Heat Equation and a Harmonic Map Flow Equation	p. 125
Nondiffusion-Type Equations	p. 129
Nonlinear Schrödinger Equations	p. 130
KdV Equation	p. 132
Notes and Comments	p. 134
A Priori Upper Bound	p. 134
Related Results on Forward Self-Similar Solutions	p. 135
Useful Analytic Tools
Various Properties of Solutions of the Heat Equation	p. 141
Convolution, the Young Inequality, and L^p-L^q Estimates	p. 141
The Young Inequality	p. 142
Proof of L^p-L^q Estimates	p. 145
Algebraic Properties of Convolution	p. 145
Interchange of Differentiation and Convolution	p. 146
Interchange of Limit and Differentiation	p. 149
Smoothness of the Solution of the Heat Equation	p. 150
Initial Values of the Heat Equation	p. 150
Convergence to the Initial Value	p. 150
Uniform Continuity	p. 151
Convergence Theorem	p. 151
Corollary	p. 153
Applications of the Convergence Theorem 4.2.3	p. 153
Inhomogeneous Heat Equations	p. 154
Representation of Solutions	p. 155
Solutions of the Inhomogeneous Equation: Case of Zero Initial Value	p. 156
Solutions of Inhomogeneous Equations: General Case	p. 160
Singular Inhomogeneous Term at t = 0	p. 160
Uniqueness of Solutions of the Heat Equation	p. 164
Proof of the Uniqueness Theorem 1.4.6	p. 164
Fundamental Uniqueness Theorem	p. 164
Inhomogeneous Equation	p. 167
Unique Solvability for Heat Equations with Transport Term	p. 168
Fundamental Solutions and Their Properties	p. 174
Integration by Parts	p. 177
An Example for Integration by Parts in the Whole Space	p. 178
A Whole Space Divergence Theorem	p. 179
Integration by Parts on Bounded Domains	p. 179
Compactness Theorems	p. 181
Compact Domains of Definition	p. 181
Ascoli-Arzelà Theorem	p. 181
Compact Embeddings	p. 184
Noncompact Domains of Definition	p. 185
Ascoli-Arzelà- Type Compactness Theorem	p. 185
Construction of Subsequences	p. 186
Equidecay and Uniform Convergence	p. 186
Proof of Lemma 1.3.6	p. 187
Convergence of Higher Derivatives	p. 187
Calculus Inequalities	p. 189
The Gagliardo-Nirenberg Inequality and the Nash Inequality	p. 189
The Gagliardo-Nirenberg Inequality	p. 190
The Nash Inequality	p. 191
Proof of the Nash Inequality	p. 191
Proof of the Gagliardo-Nirenberg Inequality (Case of ¿ < 1)	p. 194
Remarks on the Proofs	p. 199
A Remark on Assumption (6.3)	p. 199
Boundedness of the Riesz Potential	p. 200
The Hardy-Littlewood-Sobolev Inequality	p. 200
The Distribution Function and L^p-Integrability	p. 201
Lorentz Spaces	p. 203
The Marcinkiewicz Interpolation Theorem	p. 203
Gauss Kernel Representation of the Riesz Potential	p. 209
Proof of the Hardy-Littlewood-Sobolev Inequality	p. 210
Completion of the Proof	p. 212
The Sobolev Inequality	p. 212
The Inverse of the Laplacian (n ≥ 3)	p. 212
The Inverse of the Laplacian (n = 2)	p. 214
Proof of the Sobolev Inequality (r > 1)	p. 216
An Elementary Proof of the Sobolev Inequality (r = 1)	p. 217
The Newton Potential	p. 218
Remark on Differentiation Under the Integral Sign	p. 221
Boundedness of Singular Integral Operators	p. 222
Cube Decomposition	p. 222
The Calderón-Zygmund Inequality	p. 225
L² Boundedness	p. 227
Weak L¹ Estimate	p. 228
Completion of the Proof	p. 234
Notes and Comments	p. 234
Convergence Theorems in the Theory of Integration	p. 239
Interchange of Integration and Limit Operations	p. 239
Dominated Convergence Theorem	p. 240
Fatou's Lemma	p. 242
Monotone Convergence Theorem	p. 242
Convergence for Riemann Integrals	p. 243
Commutativity of Integration and Differentiation	p. 244
Differentiation Under the Integral Sign	p. 244
Commutativity of the Order of Integration	p. 245
Bounded Extension	p. 246
Answers to Exercises	p. 249
Comments of Further References	p. 273
References	p. 275
Glossary	p. 289
Index	p. 293
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Nonlinear Partial Differential Equations

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