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THE PME AS AN ABSTRACT EVOLUTION EQUATION. SEMIGROUP APPROACH

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/9780198569039.003.0010
Subject:
Mathematics, Mathematical Physics

This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation with values in ... More


THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS

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/9780198569039.003.0011
Subject:
Mathematics, Mathematical Physics

This chapter completes the investigation of previous chapters on the Dirichlet and Cauchy problems by applying the techniques to other important problems. It selects two directions, the Neumann ... More


THE CAUCHY PROBLEM WITH GROWING INITIAL DATA

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/9780198569039.003.0012
Subject:
Mathematics, Mathematical Physics

This chapter examines general conditions on the initial data under which the PME can be solved. The emphasis is on non-negative solutions and the data are locally integrable functions. Signed ... More


OPTIMAL EXISTENCE THEORY FOR NON-NEGATIVE SOLUTIONS

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/9780198569039.003.0013
Subject:
Mathematics, Mathematical Physics

This chapter studies the existence and uniqueness of solutions of the Cauchy problem for the PME posed in the whole space, which take a Radon measure as initial data. Section 13.1 constructs limit ... More


ONE-DIMENSIONAL THEORY. REGULARITY AND INTERFACES

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/9780198569039.003.0015
Subject:
Mathematics, Mathematical Physics

This chapter presents an introduction to some of the main topics relating to the PME, focusing on non-negative solutions. Section 15.1 presents a detailed analysis of the regularity of the pressure, ... More


FULL ANALYSIS OF SELF-SIMILARITY

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/9780198569039.003.0016
Subject:
Mathematics, Mathematical Physics

This chapter examines the class of solutions of the PME that are invariant under the scaling group in the variables (x, t, u), and take therefore the so-called self-similar form. After a detailed ... More


TECHNIQUES OF SYMMETRIZATION AND CONCENTRATION

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/9780198569039.003.0017
Subject:
Mathematics, Mathematical Physics

This chapter aims to obtain basic estimates for the PME and related equations using as a main tool the techniques of symmetrization and mass concentration comparison, combined with scaling ... More


ASYMPTOTIC BEHAVIOUR I. THE CAUCHY PROBLEM

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/9780198569039.003.0018
Subject:
Mathematics, Mathematical Physics

This chapter begins with a study of the behaviour of solutions of the PME for large times. The cornerstone of the presentation is the interplay between asymptotic behaviour and self-similarity. It is ... More


REGULARITY AND FINER ASYMPTOTICS IN SEVERAL DIMENSIONS

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/9780198569039.003.0019
Subject:
Mathematics, Mathematical Physics

This chapter investigates the precise regularity that is to be expected for non-negative solutions of the PME that have free boundaries. Section 19.1 describes the results of Caffarelli, Vázquez, and ... More


ASYMPTOTIC BEHAVIOUR II. DIRICHLET AND NEUMANN PROBLEMS

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/9780198569039.003.0020
Subject:
Mathematics, Mathematical Physics

This chapter contains a complete study of the large-time behaviour of solutions of the porous medium equation, ut = Δum with m > 1, posed in a bounded ... More


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