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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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- no detail
- some detail
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