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Discrete & Continuous Dynamical Systems - S

2008 , Volume 1 , Issue 3

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Eduard Feireisl, Mirko Rokyta and Josef Málek
2008, 1(3): i-iii doi: 10.3934/dcdss.2008.1.3i +[Abstract](55) +[PDF](55.7KB)
The Tenth International School on Mathematical Theory in Fluid Mechanics was held at the small village of Paseky in the northern part of the Czech Republic in May 11--18, 2007. The main part of the program of the school consisted of series of lectures delivered by Thomas Alazard, Camillo De Lellis, Eduard Feireisl, Isabelle Gallagher, and Herbert Koch. The presented book contains five survey contributions, based on the respective series of lectures.
       The article "A minicourse on the low Mach number limit" by Thomas Alazard is devoted to the study of the low Mach number limit for classical solutions of the compressible Navier-Stokes or Euler equations for non-isentropic flows. The general case is studied, in which the combined effects of large temperature variations and thermal conduction are taken into account.

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A minicourse on the low Mach number limit
Thomas Alazard
2008, 1(3): 365-404 doi: 10.3934/dcdss.2008.1.365 +[Abstract](115) +[PDF](477.4KB)
These lectures are devoted to the study of the low Mach number limit for classical solutions of the compressible Navier-Stokes or Euler equations for non-isentropic fluids. We study the general case where the combined effects of large temperature variations and thermal conduction are taken into account.
ODEs with Sobolev coefficients: The eulerian and the lagrangian approach
Camillo De Lellis
2008, 1(3): 405-426 doi: 10.3934/dcdss.2008.1.405 +[Abstract](111) +[PDF](277.6KB)
In this paper we describe two approaches to the well-posedness of Lagrangian flows of Sobolev vector fields. One is the theory of renormalized solutions which was introduced by DiPerna and Lions in the eighties. In this framework the well-posedness of the flow is a corollary of an analogous result for the corresponding transport equation. The second approach has been recently introduced by Gianluca Crippa and the author and it is instead based on suitable estimates performed directly on the lagrangian formulation.
Navier's slip and incompressible limits in domains with variable bottoms
Eduard Feireisl, Josef Málek and Antonín Novotný
2008, 1(3): 427-460 doi: 10.3934/dcdss.2008.1.427 +[Abstract](97) +[PDF](408.7KB)
We consider unsteady flows of compressible Navier-Stokes-Fourier equations in domains with bottoms that are not flat and where the fluid fulfils Navier's slip boundary conditions. Dealing with weak solutions whose long-time and large data existence has been recently established, we investigate their behavior for vanishing Mach number (the square of this small parameter appears also in the Navier slip condition), and prove their convergence towards the weak solution of the so-called Boussinesq approximation with the no-slip boundary condition. The fact that we treat the Navier boundary condition brings several interesting features in the analysis of acoustic waves, in particular the strong convergence of the velocity field.
A mathematical review of the analysis of the betaplane model and equatorial waves
Isabelle Gallagher
2008, 1(3): 461-480 doi: 10.3934/dcdss.2008.1.461 +[Abstract](79) +[PDF](254.8KB)
This review paper is devoted to the presentation of recent progress in the mathematical analysis of equatorial waves. After a short presentation of the physical background, we present some of the main mathematical results related to the problem.
    More precisely we are interested in the study of the shallow water equations set in the vicinity of the equator: in that situation the Coriolis force vanishes and its linearization near zero leads to the so-called betaplane model. Our aim is to study the asymptotics of this model in the limit of small Rossby and Froude numbers. We show in a first part the existence and uniqueness of bounded (strong) solutions on a uniform time, and we study their weak limit. In a second part we give a more precise account of the asymptotics by characterizing the possible defects of compactness to that limit, in the framework of weak solutions only.
    These results are based on the studies [6]-[8] on the one hand, and [11] on the other.
Partial differential equations with non-Euclidean geometries
Herbert Koch
2008, 1(3): 481-504 doi: 10.3934/dcdss.2008.1.481 +[Abstract](102) +[PDF](300.2KB)
Coifman and Weiss initiated the study of singular integral operators on spaces of homogeneous type. We study partial differential equations with a non-Euclidean geometry and we obtain weighted estimates using the theory of Muckenhoupt weights on related spaces of homogeneous type. Examples include squares of vector fields, elliptic operators with a drift and the Oseen operator.

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