Nonlinear Partial Differential Equations

This work serves as an excellent first course in modern analysis. The main focus 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 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.

Challenges the reader with many exercises, examples and illustrations Includes recent developments and several open problems Serves as an excellent textbook for a first course in modern analysis or as an excellent resource for a self-study guide Key topics in nonlinear partial differential equations as well as several fundamental tools and methods are presented Includes supplementary material: sn.pub/extras

Klappentext

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.

Inhalt
Asymptotic Behavior of Solutions of Partial Differential Equations.- Behavior Near Time Infinity of Solutions of the Heat Equation.- Behavior Near Time Infinity of Solutions of the Vorticity Equations.- Self-Similar Solutions for Various Equations.- Useful Analytic Tools.- Various Properties of Solutions of the Heat Equation.- Compactness Theorems.- Calculus Inequalities.- Convergence Theorems in the Theory of Integration.