The Porous Medium Equation
Mathematical Theory
Vazquez, Juan Luis Universidad Autónoma de Madrid
Print publication date: 2006 (this edition)
Published to Oxford Scholarship Online: September 2007
Print ISBN-13: 978-0-19-856903-9




2 MAIN APPLICATIONS
doi:10.1093/acprof:oso/9780198569039.003.0002

Juan Luis Vázquez
Abstract: This chapter reviews the main applications of the porous medium equation (PME). Section 2.1 describes the standard model of gas flow through a porous medium (Darcy-Leibenzon-Muskat). Section 2.2 describes the model of nonlinear heat transfer (Zel'dovich-Raizer). Section 2.3 describes Boussinesq's model of groundwater flow, and Section 2.4 presents a model of population dynamics (Gurtin-McCamy).

Keywords: PME, gas flow, nonlinear heat transfer, Boussinesq's equation of groundwater flow, population dynamics,

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Contents
 
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Full Book Contents
Preface
1 INTRODUCTION
PART ONE
2 MAIN APPLICATIONS
3 PRELIMINARIES AND BASIC ESTIMATES
4 BASIC EXAMPLES
5 THE DIRICHLET PROBLEM I. WEAK SOLUTIONS
6 THE DIRICHLET PROBLEM II. LIMIT SOLUTIONS, VERY WEAK SOLUTIONS AND SOME OTHER VARIANTS
7 CONTINUITY OF LOCAL SOLUTIONS
8 THE DIRICHLET PROBLEM III. STRONG SOLUTIONS
9 THE CAUCHY PROBLEM. L -THEORY
10 THE PME AS AN ABSTRACT EVOLUTION EQUATION. SEMIGROUP APPROACH
11 THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS
PART TWO
12 THE CAUCHY PROBLEM WITH GROWING INITIAL DATA
13 OPTIMAL EXISTENCE THEORY FOR NON-NEGATIVE SOLUTIONS
14 PROPAGATION PROPERTIES
15 ONE-DIMENSIONAL THEORY. REGULARITY AND INTERFACES
16 FULL ANALYSIS OF SELF-SIMILARITY
17 TECHNIQUES OF SYMMETRIZATION AND CONCENTRATION
18 ASYMPTOTIC BEHAVIOUR I. THE CAUCHY PROBLEM
19 REGULARITY AND FINER ASYMPTOTICS IN SEVERAL DIMENSIONS
20 ASYMPTOTIC BEHAVIOUR II. DIRICHLET AND NEUMANN PROBLEMS
COMPLEMENTS
21 FURTHER APPLICATIONS
Appendix
Bibliography
Index
 
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