Tools and Problems in Partial Differential Equations - Zuily, Claude

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Résumé :
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, Schr?dinger 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. ...

Biographie:
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 Sup?rieure and the Ecole Normale Sup?rieure 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 Universit? Paris-Sud (Orsay), where he was professor of mathematics until 2010. Currently emeritus professor at Universit? Paris-Saclay, he is the author of several books: Uniqueness and non uniqueness in the Cauchy problem (Birkh?user 1983), Probl?mes de distributions et d'?quations aux d?riv?es partielles (Hermann 1995 and Cassini 2010), Analyse pour l'agr?gation (with H. Queff?lec) (Dunod 1995), Distributions et ?quations aux d?riv?es partielles (Dunod 2002). His primary areas of research are linear and nonlinear partial differentialequations. ...

Sommaire:
Albert Ai is a postdoctoral researcher at the University of Wisconsin-Madison. He received his PhD from the University of California, Berkeley in 2019. His primary research interests include the analysis of nonlinear dispersive PDEs and harmonic analysis. In particular, he has worked on low regularity solutions of fluid models and wave equations. Thomas Alazard is Director of Research at the CNRS and Associate Professor at the ?cole normale sup?rieure Paris-Saclay. He received his PhD from the University of Bordeaux, France, in 2005. Previously, he worked at the Orsay mathematics department and at the ?cole normale sup?rieure in Paris. His research focuses on the analysis of partial differential equations, a subject on which he has written several books. Mihaela Ifrim is currently a Professor of Mathematics at the University of Wisconsin, Madison. She studied mathematics in Bucharest and at the University of California, Davis, where she received her Ph.D. in 2012. After a Simons postdoctoral fellowship at UC Berkeley, in 2017 she moved Madison, where she was a Sloan Research Fellow and a CAREER grant recipient. Her work spans many directions in nonlinear partial differential equations, including fluid dynamics and nonlinear dispersive flows, with an emphasis on free boundary problems and on the study of low regularity local and global dynamics of the solutions. Daniel Tataru is a Distinguished Professor in Mathematics at the University of California, Berkeley. He is well known for his substantial contributions to dispersive pde's, also in connection to fluid dynamics, harmonic analysis, geometry, general relativity and free boundary problems. He studied mathematics at the University of Iasi, in Romania, and then at University of Virginia, where he received his Ph.D. in 1992. He spent almost a decade at Northwestern university, before moving to Berkeley. Among other honors, he is a Fellow of both the American Academy of Arts and Sciences and the European Academy of Sciences, as well as a Simons Investigator since 2013....