This textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations.Through 65 fully solved problems, the book offers readers a fast but in-depth introduction to the field, covering advanced topics in microlocal analysis, including pseudo- and para-differential calculus, and the key classical equations, such as the Laplace, Schrödinger or Navier-Stokes equations. Essentially self-contained, the book begins with problems on the necessary tools from functional analysis, distributions, and the theory of functional spaces, and in each chapter the problems are preceded by a summary of the relevant results of the theory.

Informed by the authors' extensive research experience and years of teaching, this book is for graduate students and researchers who wish to gain real working knowledge of the subject.

Autorentext

Thomas Alazard is a senior researcher at CNRS. For several years, he has taught partial differential equations, microlocal analysis and functional analysis at the Ecole Normale Supérieure and the Ecole Normale Supérieure Paris-Saclay. His research focuses on the applications of harmonic analysis and microlocal analysis to the study of nonlinear partial differential equations.Claude Zuily received his PhD from Université Paris-Sud (Orsay), where he was professor of mathematics until 2010. Currently emeritus professor at Université Paris-Saclay, he is the author of several books: Uniqueness and non uniqueness in the Cauchy problem (Birkhäuser 1983), Problèmes de distributions et d'équations aux dérivées partielles (Hermann 1995 and Cassini 2010), Analyse pour l'agrégation (with H. Queffélec) (Dunod 1995), Distributions et équations aux dérivées partielles (Dunod 2002). His primary areas of research are linear and nonlinear partial differential equations.

Inhalt

Part I Tools and Problems.- 1 Elements of functional analysis and distributions.- 2 Statements of the problems of Chapter 1.- 3 Functional spaces.- 4 Statements of the problems of Chapter 3.- 5 Microlocal analysis.- 6 Statements of the problems of Chapter 5.- 7 The classical equations.- 8 Statements of the problems of Chapter 7.- Part II Solutions of the Problems. A Classical results. Index.