Nonlinear Partial Differential Equations

1. Front Matter

   Pages i-xviii

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2. Asymptotic Behavior of Solutions of Partial Differential Equations

   1. Front Matter

      Pages 1-1

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   2. Behavior Near Time Infinity of Solutions of the Heat Equation

      • Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

      Pages 3-36

   3. Behavior Near Time Infinity of Solutions of the Vorticity Equations

      • Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

      Pages 37-103

   4. Self-Similar Solutions for Various Equations

      • Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

      Pages 105-138

3. Useful Analytic Tools

   1. Front Matter

      Pages 140-140

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4. Useful Analytic Tools

   1. Various Properties of Solutions of the Heat Equation

      • Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

      Pages 141-180

   2. Compactness Theorems

      • Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

      Pages 181-188

   3. Calculus Inequalities

      • Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

      Pages 189-238

   4. Convergence Theorems in the Theory of Integration

      • Mi-Ho Giga, Yoshikazu Giga, Jürgen Saal

      Pages 239-247

5. Back Matter

   Pages 249-294

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The purpose of this book is to present typical methods (including rescaling methods) for the examination of the behavior of solutions of nonlinear partial di?erential equations of di?usion type. For instance, we examine such eq- tions by analyzing special so-called self-similar solutions. We are in particular interested in equations describing various phenomena such as the Navier– Stokesequations.Therescalingmethod describedherecanalsobeinterpreted as a renormalization group method, which represents a strong tool in the asymptotic analysis of solutions of nonlinear partial di?erential equations. Although such asymptotic analysis is used formally in various disciplines, not seldom there is a lack of a rigorous mathematical treatment. The intention of this monograph is to ?ll this gap. We intend to develop a rigorous mat- matical foundation of such a formalasymptotic analysis related to self-similar solutions. A self-similar solution is, roughly speaking, a solution invariant under a scaling transformationthat does not change the equation. For several typical equations we shall give mathematical proofs that certain self-similar solutions asymptotically approximate the typical behavior of a wide class of solutions. Since nonlinear partial di?erential equations are used not only in mat- matics but also in various ?elds of science and technology, there is a huge variety of approaches. Moreover,even the attempt to cover only a few typical ?elds and methods requires many pages of explanations and collateral tools so that the approaches are self-contained and accessible to a large audience.

• Burgers vortex
• Navier-Stokes equations
• asymptotic behavior
• calculus
• calculus inequalities
• compactness
• differential equation
• mean curvature flow
• mean curvature flow equations
• nonlinear functional analysis
• nonlinear partial differential equations;
• partial differential equations

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.” (Jesús Hernández, Mathematical Reviews, Issue 2011 f)

“The book presents typical methods … for the examination of the behavior of solutions of nonlinear partial differential equations of diffusion type. … The aim of the authors was to teach the readers to deal with such tools during the study of PDEs and to give them a strong motivation for their study. … The book is written in a very pedagogical way. Each chapter contains plenty of exercises whose detailed solutions can be found at the end of the book.” (Pavol Quittner, Zentralblatt MATH, Vol. 1215, 2011)

• , Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Japan

  Mi-Ho Giga

• Graduate School of, Mathematical Sciences, University of Tokyo, Tokyo, Japan

  Yoshikazu Giga

• , Center of Smart Interfaces, Technische Universität Darmstadt, Darmstadt, Germany

  Jürgen Saal

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