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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


FURTHER APPLICATIONS

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/9780198569039.003.0021
Subject:
Mathematics, Mathematical Physics

This chapter presents a collection of new examples taken from different branches of science. The first examples come from fluid dynamics, starting with the well-known model of viscous droplets ... More


MAIN APPLICATIONS

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/9780198569039.003.0002
Subject:
Mathematics, Mathematical Physics

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 ... More


PRELIMINARIES AND BASIC ESTIMATES

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/9780198569039.003.0003
Subject:
Mathematics, Mathematical Physics

This chapter provides preliminary material on parabolic equations needed to develop the main theories of the book. It begins with a review of useful properties of quasilinear parabolic equations. ... More


THE DIRICHLET PROBLEM I. WEAK 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/9780198569039.003.0005
Subject:
Mathematics, Mathematical Physics

This chapter begins with a systematic study of the questions of existence, uniqueness, and main properties of the solutions of the PME by concentrating on the first boundary-value problem posed in a ... More


THE DIRICHLET PROBLEM III. STRONG 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/9780198569039.003.0008
Subject:
Mathematics, Mathematical Physics

This chapter addresses the question of how regular the solutions constructed in previous chapters actually are. Section 8.1 considers the question of further regularity of the time derivative ut ... More


THE CAUCHY PROBLEM. L 1-THEORY

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/9780198569039.003.0009
Subject:
Mathematics, Mathematical Physics

This chapter examines the Cauchy problem, or pure initial value problem, for the PME and the GPME in d-dimensional space, d = 1. It concentrates the main effort on the PME with zero forcing term. It ... More


Preliminaries

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/9780199202973.003.0002
Subject:
Mathematics, Applied Mathematics

This chapter provides necessary preliminaries on functional analysis, comparison results, and the fundamentals of the PME. Section 1.2 provides the basic information on the equation that will serve ... More


Smoothing effect and time decay. Data in L 1 (R n ) or M (R n )

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/9780199202973.003.0003
Subject:
Mathematics, Applied Mathematics

This chapter discusses the smoothing and decay effects for the porous medium equation, using as a model case the famous Barenblatt solutions that have explicit formulas. Topics covered include ... 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


Private Equity

Andrew Ang

in Asset Management: A Systematic Approach to Factor Investing

Published in print:
2014
Published Online:
August 2014
ISBN:
9780199959327
eISBN:
9780199382323
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199959327.003.0018
Subject:
Economics and Finance, Financial Economics

Private equity (PE) is not an asset class. Performance measurement of PE is hampered because commonly used metrics are not returns and are often manipulated. On the whole, PE does not outperform ... More


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