Hardcover
Published: 1st June 2010
ISBN: 9780817641733
Number Of Pages: 294
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)
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-ArzelÃ -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}-L^{1} 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 L^{1}-Norm | p. 49 |
Application of the Nash Inequality | p. 50 |
Proof of Fundamental L^{q}-L^{1} Estimates | p. 53 |
Extension of Fundamental L^{q}-L^{1} 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 SchrÃ¶dinger 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-ArzelÃ Theorem | p. 181 |
Compact Embeddings | p. 184 |
Noncompact Domains of Definition | p. 185 |
Ascoli-ArzelÃ - 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 CalderÃ³n-Zygmund Inequality | p. 225 |
L^{2} Boundedness | p. 227 |
Weak L^{1} 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 |
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
Publisher: BIRKHAUSER BOSTON INC
Country of Publication: US
Dimensions (cm): 23.5 x 15.5
x 2.54
Weight (kg): 1.36