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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


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


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


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


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


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


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


Main Theorem

Sergiu Klainerman and Jérémie Szeftel

in Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations: (AMS-210)

Published in print:
2020
Published Online:
May 2021
ISBN:
9780691212425
eISBN:
9780691218526
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691212425.003.0003
Subject:
Mathematics, Geometry / Topology

This chapter presents the main theorem, its main conclusions, as well as a full strategy of its proof, divided in nine supporting intermediate results, Theorems M0–M8. The chapter specifies the ... 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


Holder Estimates for Higher Dimensional Corner 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/9780691157122.003.0007
Subject:
Mathematics, Probability / Statistics

This chapter establishes Hölder space estimates for higher dimensional corner model problems. It first explains the homogeneous Cauchy problem before estimating the solution of the inhomogeneous ... More


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