## Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

*Sylvie Benzoni-Gavage and Denis Serre*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199211234
- eISBN:
- 9780191705700
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199211234.001.0001
- Subject:
- Mathematics, Applied Mathematics

This book presents a view of the state of the art in multi-dimensional hyperbolic partial differential equations, with a particular emphasis on problems in which modern tools of analysis have proved ... 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

## 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

## 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

## 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

## Local Cauchy Problem

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0006
- Subject:
- Mathematics, Applied Mathematics

This chapter begins with a discussion of moving frame formulae. It then covers n + 1 splitting adapted to space slices, constraints and evolution, Hamiltonian and symplectic formulation, Cauchy ... 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

## A Review of DDE Methods

*Alfredo Bellen and Marino Zennaro*

### in Numerical Methods for Delay Differential Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506546
- eISBN:
- 9780191709609
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506546.003.0003
- Subject:
- Mathematics, Numerical Analysis

This chapter starts with a brief historical excursus of the early methods used for the numerical solution of DDEs. It then considers general formulation and convergence results for discrete and ... 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

## Non‐Linear Hyperbolic Rigid Heat Conductor of The Coleman Type

*Józef Ignaczak and Martin Ostoja‐Starzewski*

### in Thermoelasticity with Finite Wave Speeds

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199541645
- eISBN:
- 9780191716164
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199541645.003.0013
- Subject:
- Mathematics, Applied Mathematics, Mathematical Physics

While all previous chapters are devoted to linear hyperbolic theories of thermoelasticity, this one concerns a rigid but nonlinear hyperbolic heat conductor due to Coleman et al. (1982, 1983, 1986). ... More

## Looking at the World Sideways

*Craig Callender*

### in What Makes Time Special?

- Published in print:
- 2017
- Published Online:
- July 2017
- ISBN:
- 9780198797302
- eISBN:
- 9780191839603
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797302.003.0008
- Subject:
- Philosophy, Metaphysics/Epistemology

When physics tells its story of the world, it writes on spatial pages and we flip pages in the temporal directions. The present moment contains the seeds of what happens next. Relativity challenges ... More

## Stability Analysis of Runge-Kutta Methods For DDEs

*Alfredo Bellen and Marino Zennaro*

### in Numerical Methods for Delay Differential Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506546
- eISBN:
- 9780191709609
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506546.003.0010
- Subject:
- Mathematics, Numerical Analysis

This chapter addresses the stability analysis of numerical methods, mainly Runge-Kutta, with respect to the test equations introduced in Chapter 9. Due to the many-sided stability requirements and ... More

## Hamilton–Jacobi Theory

*Oliver Johns*

### in Analytical Mechanics for Relativity and Quantum Mechanics

- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567264
- eISBN:
- 9780191717987
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567264.003.0019
- Subject:
- Physics, Atomic, Laser, and Optical Physics

The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. ... More

## The Cauchy problem in general relativity

*Hans Ringström*

### in On the Topology and Future Stability of the Universe

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680290
- eISBN:
- 9780191760235
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680290.003.0002
- Subject:
- Mathematics, Mathematical Physics, Geometry / Topology

This chapter contains a description of the Cauchy problem in the general theory of relativity. The intended audience consists of physicists with basic knowledge of general relativity. Beyond the ... More

## On the Topology and Future Stability of the Universe

*Hans Ringström*

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680290
- eISBN:
- 9780191760235
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680290.001.0001
- Subject:
- Mathematics, Mathematical Physics, Geometry / Topology

The subject of the book is the topology and future stability of models of the universe. In standard cosmology, the universe is assumed to be spatially homogeneous and isotropic. However, it is of ... More

## Outline, general theory of the Einstein–Vlasov system

*Hans Ringström*

### in On the Topology and Future Stability of the Universe

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680290
- eISBN:
- 9780191760235
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680290.003.0008
- Subject:
- Mathematics, Mathematical Physics, Geometry / Topology

One important part of the book is to establish that it is meaningful to formulate a Cauchy problem for the Einstein–Vlasov equations. In Chapter 8 of the book, we give an overview of the part of the ... More

## Main results

*Hans Ringström*

### in On the Topology and Future Stability of the Universe

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680290
- eISBN:
- 9780191760235
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680290.003.0007
- Subject:
- Mathematics, Mathematical Physics, Geometry / Topology

Chapter 7 contains a mathematical description of the main results: future stability of models of the universe and the existence of models, consistent with the observations, with topology different ... More

## Holder Estimates for Euclidean 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/9780691157122.003.0008
- Subject:
- Mathematics, Probability / Statistics

This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator ... More

## Existence of Solutions

*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/9780691157122.003.0010
- Subject:
- Mathematics, Probability / Statistics

This chapter proves existence of solutions to the inhomogeneous problem using the Schauder estimate and analyzes a generalized Kimura diffusion operator, L, defined on a manifold with corners, P. The ... 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