## 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