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Nonlinear Partial Differential Equations : Asymptotic Behavior of Solutions and Self-Similar Solutions - Yoshikazu Giga

Nonlinear Partial Differential Equations

Asymptotic Behavior of Solutions and Self-Similar Solutions

Hardcover Published: 1st June 2010
ISBN: 9780817641733
Number Of Pages: 294

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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.

Nonlinear Partial Differential Equations will serve as an excellent textbook for a first course in modern analysis or as a useful self-study guide. Key topics in nonlinear partial differential equations as well as several fundamental tools and methods are presented. The only prerequisite required is a basic course in calculus.

From the reviews: "This book studies the asymptotic behavior of solutions to some nonlinear evolution problems by using rescaling ... methods with self-similar solutions ... . not only are there exercises but also answers to these exercises. In any case this book is a very welcome and useful addition to the literature." (Jesus Hernandez, Mathematical Reviews, Issue 2011 f)

Prefacep. xiii
Asymptotic Behavior of Solutions of Partial Differential Equations
Behavior near Time Infinity of Solutions of the Heat Equationp. 3
Asymptotic Behavior of Solutions near Time Infinityp. 3
Decay Estimate of Solutionsp. 6
Lp-Lq Estimatesp. 8
Derivative Lp-Lq Estimatesp. 8
Theorem on Asymptotic Behavior Near Time Infinityp. 10
Proof Using Representation Formula of Solutionsp. 11
Integral Form of the Mean Value Theoremp. 12
Structure of Equations and Self-Similar Solutionsp. 13
Invariance Under Scalingp. 13
Conserved Quantity for the Heat Equationp. 14
Scaling Transformation Preserving the Conserved Quantityp. 15
Summary of Properties of a Scaling Transformationp. 15
Self-Similar Solutionsp. 16
Expression of Asymptotic Formula Using Scaling Transformationsp. 16
Idea of the Proof Based on Scaling Transformationp. 17
Compactnessp. 18
Family of Functions Consisting of Continuous Functionsp. 19
Ascoli-Arzelà-type Compactness Theoremp. 22
Relative Compactness of a Family of Scaled Functionsp. 22
Decay Estimates in Space Variablesp. 25
Existence of Convergent Subsequencesp. 26
Lemmap. 27
Characterization of Limit Functionsp. 27
Limit of the Initial Datap. 28
Weak Form of the Initial Value Problem for the Heat Equationp. 29
Weak Solutions for the Initial Value Problemp. 30
Limit of a Sequence of Solutions to the Heat Equationp. 31
Characterization of the Limit of a Family of Scaled Functionsp. 32
Uniqueness Theorem When Initial Data is the Delta Functionp. 33
Completion of the Proof of Asymptotic Formula (1 9) Based on Scaling Transformationp. 34
Remark on Uniqueness Theoremp. 34
Behavior Near Time Infinity of Solutions of the Vorticity Equationsp. 37
Navier-Stokes Equations and Vorticity Equationsp. 38
Vorticityp. 39
Vorticity and Velocityp. 40
Biot-Savart Lawp. 41
Derivation of the Vorticity Equationsp. 42
Asymptotic Behavior Near Time Infinityp. 42
Unique Existence Theoremp. 43
Theorem for Asymptotic Behavior of the Vorticityp. 44
Scaling Invariancep. 44
Conservation of the Total Circulationp. 45
Rotationally Symmetric Self-Similar Solutionsp. 46
Global Lq-L1 Estimates for Solutions of the Heat Equation with a Transport Termp. 47
Fundamental Lq-Lr Estimatesp. 47
Change Ratio of Lr-Norm per Time: Integral Identitiesp. 48
Nonincrease of L1-Normp. 49
Application of the Nash Inequalityp. 50
Proof of Fundamental Lq-L1 Estimatesp. 53
Extension of Fundamental Lq-L1 Estimatesp. 55
Maximum Principlep. 55
Preservation of Nonnegativityp. 56
Estimates for Solutions of Vorticity Equationsp. 58
Estimates for Vorticity and Velocityp. 58
Estimates for Derivatives of the Vorticityp. 62
Decay Estimates for the Vorticity in Spatial Variablesp. 68
Proof of the Asymptotic Formulap. 72
Characterization of the Limit Function as a Weak Solutionp. 73
Estimates for the Limit Functionp. 76
Integral Equation Satisfied by Weak Solutionsp. 80
Uniqueness of Solutions of Limit Equationsp. 81
Completion of the Proof of the Asymptotic Formulap. 83
Formation of the Burgers Vortexp. 84
Convergence to the Burgers Vortexp. 85
Asymmetric Burgers Vorticesp. 87
Self-Similar Solutions of the Navier-Stokes Equations and Related Topicsp. 88
Short History of Research on Asymptotic Behavior of Vorticityp. 89
Problems of Existence of Solutionsp. 91
Self-Similar Solutionsp. 93
Uniqueness of the Limit Equation for Large Circulationp. 97
Uniqueness of Weak Solutionsp. 97
Relative Entropyp. 98
Boundedness of the Entropyp. 100
Rescalingp. 100
Proof of the Uniqueness Theoremp. 101
Remark on Asymptotic Behavior of the Vorticityp. 102
Self-Similar Solutions for Various Equationsp. 105
Porous Medium Equationp. 105
Self-Similar Solutions Preserving Total Massp. 107
Weak Solutionsp. 108
Asymptotic Formulap. 109
Roles of Backward Self-Similar Solutionsp. 109
Axisymmetric Mean Curvature Flow Equationp. 110
Backward Self-Similar Solutions and Similarity Variablesp. 111
Nonexistence of Nontrivial Self-Similar Solutionsp. 114
Asymptotic Behavior of Solutions Near Pinching Pointsp. 116
Monotonicity Formulap. 121
The Cases of a Semilinear Heat Equation and a Harmonic Map Flow Equationp. 125
Nondiffusion-Type Equationsp. 129
Nonlinear Schrödinger Equationsp. 130
KdV Equationp. 132
Notes and Commentsp. 134
A Priori Upper Boundp. 134
Related Results on Forward Self-Similar Solutionsp. 135
Useful Analytic Tools
Various Properties of Solutions of the Heat Equationp. 141
Convolution, the Young Inequality, and Lp-Lq Estimatesp. 141
The Young Inequalityp. 142
Proof of Lp-Lq Estimatesp. 145
Algebraic Properties of Convolutionp. 145
Interchange of Differentiation and Convolutionp. 146
Interchange of Limit and Differentiationp. 149
Smoothness of the Solution of the Heat Equationp. 150
Initial Values of the Heat Equationp. 150
Convergence to the Initial Valuep. 150
Uniform Continuityp. 151
Convergence Theoremp. 151
Corollaryp. 153
Applications of the Convergence Theorem 4.2.3p. 153
Inhomogeneous Heat Equationsp. 154
Representation of Solutionsp. 155
Solutions of the Inhomogeneous Equation: Case of Zero Initial Valuep. 156
Solutions of Inhomogeneous Equations: General Casep. 160
Singular Inhomogeneous Term at t = 0p. 160
Uniqueness of Solutions of the Heat Equationp. 164
Proof of the Uniqueness Theorem 1.4.6p. 164
Fundamental Uniqueness Theoremp. 164
Inhomogeneous Equationp. 167
Unique Solvability for Heat Equations with Transport Termp. 168
Fundamental Solutions and Their Propertiesp. 174
Integration by Partsp. 177
An Example for Integration by Parts in the Whole Spacep. 178
A Whole Space Divergence Theoremp. 179
Integration by Parts on Bounded Domainsp. 179
Compactness Theoremsp. 181
Compact Domains of Definitionp. 181
Ascoli-Arzelà Theoremp. 181
Compact Embeddingsp. 184
Noncompact Domains of Definitionp. 185
Ascoli-Arzelà- Type Compactness Theoremp. 185
Construction of Subsequencesp. 186
Equidecay and Uniform Convergencep. 186
Proof of Lemma 1.3.6p. 187
Convergence of Higher Derivativesp. 187
Calculus Inequalitiesp. 189
The Gagliardo-Nirenberg Inequality and the Nash Inequalityp. 189
The Gagliardo-Nirenberg Inequalityp. 190
The Nash Inequalityp. 191
Proof of the Nash Inequalityp. 191
Proof of the Gagliardo-Nirenberg Inequality (Case of < 1)p. 194
Remarks on the Proofsp. 199
A Remark on Assumption (6.3)p. 199
Boundedness of the Riesz Potentialp. 200
The Hardy-Littlewood-Sobolev Inequalityp. 200
The Distribution Function and Lp-Integrabilityp. 201
Lorentz Spacesp. 203
The Marcinkiewicz Interpolation Theoremp. 203
Gauss Kernel Representation of the Riesz Potentialp. 209
Proof of the Hardy-Littlewood-Sobolev Inequalityp. 210
Completion of the Proofp. 212
The Sobolev Inequalityp. 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 Potentialp. 218
Remark on Differentiation Under the Integral Signp. 221
Boundedness of Singular Integral Operatorsp. 222
Cube Decompositionp. 222
The Calderón-Zygmund Inequalityp. 225
L2 Boundednessp. 227
Weak L1 Estimatep. 228
Completion of the Proofp. 234
Notes and Commentsp. 234
Convergence Theorems in the Theory of Integrationp. 239
Interchange of Integration and Limit Operationsp. 239
Dominated Convergence Theoremp. 240
Fatou's Lemmap. 242
Monotone Convergence Theoremp. 242
Convergence for Riemann Integralsp. 243
Commutativity of Integration and Differentiationp. 244
Differentiation Under the Integral Signp. 244
Commutativity of the Order of Integrationp. 245
Bounded Extensionp. 246
Answers to Exercisesp. 249
Comments of Further Referencesp. 273
Referencesp. 275
Glossaryp. 289
Indexp. 293
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780817641733
ISBN-10: 0817641734
Series: Progress in Nonlinear Differential Equations and Their Applications : Book 79
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 294
Published: 1st June 2010
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 0.62