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Dynamics of Viscous Compressible Fluids

Eduard Feireisl

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198528388
eISBN:
9780191713590
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528388.001.0001
Subject:
Mathematics, Applied Mathematics

The book presents and develops the most recent ideas and concepts of the mathematical theory of viscous, compressible, and heat conducting fluids. Two main objectives are pursued: (i) global ... 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 II. LIMIT SOLUTIONS, VERY WEAK SOLUTIONS AND SOME OTHER VARIANTS

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/9780198569039.003.0006
Subject:
Mathematics, Mathematical Physics

This chapter continues the analysis of the initial and boundary value problem. Topics covered include limit solutions, theory of very weak solutions, problems in different domains, limit solutions to ... 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


VARIATIONAL SOLUTIONS

Eduard Feireisl

in Dynamics of Viscous Compressible Fluids

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198528388
eISBN:
9780191713590
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528388.003.0004
Subject:
Mathematics, Applied Mathematics

The concept of a variational solution to the full Navier-Stokes-Fourier system is introduced in this chapter. Each equation of the complete system, that means, the equation of continuity, the ... More


MATHEMATICAL ANALYSIS OF TWO-FLUID PROBLEMS

Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre

in Mathematical Methods for the Magnetohydrodynamics of Liquid Metals

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198566656
eISBN:
9780191718014
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566656.003.0004
Subject:
Mathematics, Mathematical Physics

This chapter deals with the theoretical aspects of multifluid magnetohydrodynamics problems. In addition to the coupling between hydrodynamics and electromagnetics examined in Chapter 2, the high ... More


CONTINUITY OF LOCAL 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/9780198569039.003.0007
Subject:
Mathematics, Mathematical Physics

This chapter addresses a main issue of the theory, namely, the continuity of the solutions for times t > 0. The equation treated in this chapter is a generalized version of the GPME. The question of ... More


Perturbative weak KAM theory

Kaloshin Vadim and Zhang Ke

in Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom: (AMS-208)

Published in print:
2020
Published Online:
May 2021
ISBN:
9780691202525
eISBN:
9780691204932
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691202525.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter explores perturbation aspects of the weak Kolmogorov-Arnold-Moser (KAM) theory. By perturbative weak KAM theory, we mean two things. How do the weak KAM solutions and the Mather, Aubry, ... More


Stochastic Differential Equations

Gopinath Kallianpur and P. Sundar

in Stochastic Analysis and Diffusion Processes

Published in print:
2014
Published Online:
April 2014
ISBN:
9780199657063
eISBN:
9780191781759
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199657063.003.0006
Subject:
Mathematics, Probability / Statistics, Applied Mathematics

Stochastic differential equations arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of ... More


The Euler-Reynolds System

Philip Isett

in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691174822
eISBN:
9781400885428
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691174822.003.0001
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter provides a background on the Euler-Reynolds system, starting with some of the underlying philosophy behind the argument. It describes low frequency parts and ensemble averages of Euler ... More


Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Philip Isett

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691174822
eISBN:
9781400885428
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691174822.001.0001
Subject:
Mathematics, Computational Mathematics / Optimization

Motivated by the theory of turbulence in fluids, the physicist and chemist Lars Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations might fail to conserve energy if ... More


Structure of the Book

Philip Isett

in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691174822
eISBN:
9781400885428
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691174822.003.0002
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter provides an overview of the book's structure. Section 3 deals with the error terms which need to be controlled, whereas Part III explains some notation of the book and presents a basic ... More


Frequency and Energy Levels

Philip Isett

in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691174822
eISBN:
9781400885428
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691174822.003.0009
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter shows how to measure the Hölder regularity of the weak solutions that are constructed when the scheme is executed more carefully. For this aspect of the convex integration scheme, a ... More


On Onsager's Conjecture

Philip Isett

in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691174822
eISBN:
9781400885428
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691174822.003.0013
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter deals with Onsager's conjecture, which would be implied by a stronger form of Lemma (10.1). It considers what could be proven assuming Conjecture (10.1) by turning to Theorem 13.1, which ... More


Dispersive Cases

Jean-Yves Chemin, Benoit Desjardins, Isabelle Gallagher, and Emmanuel Grenier

in Mathematical Geophysics: An introduction to rotating fluids and the Navier-Stokes equations

Published in print:
2006
Published Online:
November 2020
ISBN:
9780198571339
eISBN:
9780191916748
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198571339.003.0011
Subject:
Earth Sciences and Geography, Geophysics: Earth Sciences

It is well known that dispersive phenomena play a significant role in the study of partial differential equations. Historically, the use of dispersive ... More


Rotating Shallow-Water Models as Quasilinear Hyperbolic Systems, and Related Numerical Methods

Vladimir Zeitlin

in Geophysical Fluid Dynamics: Understanding (almost) everything with rotating shallow water models

Published in print:
2018
Published Online:
April 2018
ISBN:
9780198804338
eISBN:
9780191842627
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198804338.003.0007
Subject:
Physics, Geophysics, Atmospheric and Environmental Physics

The chapter contains the mathematical background necessary to understand the properties of RSW models and numerical methods for their simulations. Mathematics of RSW model is presented by using their ... More


Weak Solutions of the Navier–Stokes Equations

Jean-Yves Chemin, Benoit Desjardins, Isabelle Gallagher, and Emmanuel Grenier

in Mathematical Geophysics: An introduction to rotating fluids and the Navier-Stokes equations

Published in print:
2006
Published Online:
November 2020
ISBN:
9780198571339
eISBN:
9780191916748
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198571339.003.0007
Subject:
Earth Sciences and Geography, Geophysics: Earth Sciences

The mathematical analysis of the incompressible Stokes and Navier–Stokes equations in a possibly unbounded domain Ω of Rd (d = 2 or 3) is the purpose of ... More


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