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The Porous Medium Equation: Mathematical Theory

Juan Luis Vazquez

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198569039
eISBN:
9780191717468
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198569039.001.0001
Subject:
Mathematics, Mathematical Physics

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 ... More


Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

Juan Luis Vázquez

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199202973
eISBN:
9780191707919
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199202973.001.0001
Subject:
Mathematics, Applied Mathematics

This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as ... More


INTRODUCTION

Juan Luis Vázquez

in The Porous Medium Equation: Mathematical Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198569039
eISBN:
9780191717468
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198569039.003.0001
Subject:
Mathematics, Mathematical Physics

This chapter introduces the mathematical theory of the nonlinear heat equation referred to as the porous medium equation (PME). This equation is one of the simplest examples of a nonlinear evolution ... More


Lower bounds, contractivity, error estimates, and continuity

Juan Luis Vázquez

in Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199202973
eISBN:
9780191707919
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199202973.003.0005
Subject:
Mathematics, Applied Mathematics

This chapter considers a number of bounds that complement the study so far, which was centred on upper bounds. The first section concerns the existence of lower bounds for non-negative solutions. ... More


PROPAGATION PROPERTIES

Juan Luis Vázquez

in The Porous Medium Equation: Mathematical Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198569039
eISBN:
9780191717468
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198569039.003.0014
Subject:
Mathematics, Mathematical Physics

This chapter introduces a study of the properties of the support and the free boundary of the solution in a several dimensional setting. It discusses in detail the property of finite propagation and ... More


EFFECT OF FLUCTUATIONS ON THERMOELECTRICITY AND HEAT TRANSPORT

Anatoly Larkin and Andrei Varlamov

in Theory of Fluctuations in Superconductors

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198528159
eISBN:
9780191713521
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528159.003.0010
Subject:
Physics, Condensed Matter Physics / Materials

This chapter introduces a phenomenological definition of the heat current. An explicit expression for the heat current operator in interacting electron system is derived in the framework of ... More


Parabolic Equations

Rüdiger Verfürth

in A Posteriori Error Estimation Techniques for Finite Element Methods

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199679423
eISBN:
9780191758485
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199679423.003.0006
Subject:
Mathematics, Applied Mathematics, Numerical Analysis

We derive a posteriori error estimates for space–time finite element and certain finite volume discretisations of linear and nonlinear parabolic equations of second order. The starting point of our ... More


Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Charles L. Epstein and Rafe Mazzeo

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157122
eISBN:
9781400846108
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157122.001.0001
Subject:
Mathematics, Probability / Statistics

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as ... More


Two Contrasting Characters: Fourier and Herapath

Jennifer Coopersmith

in Energy, the Subtle Concept: The discovery of Feynman's blocks from Leibniz to Einstein

Published in print:
2015
Published Online:
August 2015
ISBN:
9780198716747
eISBN:
9780191800955
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198716747.003.0011
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, History of Physics

Herapath was an eccentric Englishman who developed a kinetic theory of gases. He was the first to estimate the typical speed of a gas molecule. His paper was rejected by Humphry Davy, and Herapath ... More


The Resolvent Operator

Charles L. Epstein and Rafe1 Mazzeo

in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157122
eISBN:
9781400846108
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157122.003.0011
Subject:
Mathematics, Probability / Statistics

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) ... More


Maximum Principles and Uniqueness Theorems

Charles L. Epstein and Rafe1 Mazzeo

in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157122
eISBN:
9781400846108
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157122.003.0003
Subject:
Mathematics, Probability / Statistics

This chapter proves maximum principles for two parabolic and elliptic equations from which the uniqueness results follow easily. It also considers the main consequences of the maximum principle, both ... More


The Model Solution Operators

Charles L. Epstein and Rafe1 Mazzeo

in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157122
eISBN:
9781400846108
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157122.003.0004
Subject:
Mathematics, Probability / Statistics

This chapter introduces the model problems and the solution operator for the associated heat equations. These operators give a good approximation for the behavior of the heat kernel in neighborhoods ... More


Holder Estimates for General Models

Charles L. Epstein and Rafe1 Mazzeo

in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691157122
eISBN:
9781400846108
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691157122.003.0009
Subject:
Mathematics, Probability / Statistics

This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining ... More


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