The Porous Medium Equation: Mathematical Theory

Hardcover | October 26, 2006

byJuan Luis Vazquez

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The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academicsin mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physicalapplications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes,providing comments, historical notes or recommended reading, and exercises for the reader.

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The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academicsin mathematics and engineering, as well as e...

Juan Luis Vazquez is at Universidad Autonoma de Madrid.

other books by Juan Luis Vazquez

Format:HardcoverDimensions:648 pages, 9.21 × 6.14 × 1.57 inPublished:October 26, 2006Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198569033

ISBN - 13:9780198569039

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Table of Contents

Preface1. IntroductionPart 12. Main applications3. Preliminaries and basic estimates4. Basic examples5. The Dirichlet problem I. Weak solutions6. The Dirichlet problem II. Limit solutions, very weak solutions and some other variants7. Continuity of local solutions8. The Dirichlet problem III. Strong solutions9. The Cauchy problem. L' theory10. The PME as an abstract evolution equation. Semigroup approach11. The Neumann problem and problems on manifoldsPart 212. The Cauchy problem with growing initial data13. Optimal existence theory for nonnegative solutions14. Propagation properties15. One-dimensional theory. Regularity and interfaces16. Full analysis of selfsimilarity17. Techniques of symmetrization and concentration18. Asymptotic behaviour I. The Cauchy problem19. Regularity and finer asymptotics in several dimensions20. Asymptotic behaviour II. Dirichlet and Neumann problemsComplements21. Further applications22. Basic facts and appendicesBibliographyIndex

Editorial Reviews

"I deeply believe that this book is one of the most important works in its field that have appeared until now. It is beautifully written and well organized, and I strongly recommend it to anyone seeking a stylish, balanced, up-to-date survey of this central area of nonlinear partial differential equations." -- Vicentiu Radulescu, Mathematical Reviews