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ModTerCom Workshop: Analyse mathématique et numérique des équations de Navier-Stokes compressibles

The Workshop takes place at the IGESA of Porquerolles Island (Hyères) from 13.05.2015 to 16.05.2015.

The underlying general idea behind the MODTERCOM project is to ecourage the real scientific collaboration and exchange between specialists in numlerical analysis on one hand and analysts of PDEs on the other hand, in the field of modeling of (viscous) compressible fluids. Its ultimate aim is to obtain qualitatively new results in numerical analysis, to get qualitatively new information on the precison of existing numerical approaches and to suggest new more accurate and applicable methodologies, by employing and adapting to the dicrete level the recent advanced tools in the the theory of non linear PDEs. This workshop is one of the steps in this direction: it will help to go ahead in this programme and to get closer to this goal.

Main speakers: C. Cancès, B. Dubruelle, R. Danchin, V. Estella-Perez, R. Eymard, D. Faranda, D. Gérard-Varet, B. Haspot, Y. Robin, N. Therme, S. Vannitsen, M. H. Vignal.

Orgenizers: R. Herbin, T. Gallouet (I2M, Aix Marseille Université), A. Novotny (IMATH, Université de Toulon).

Oberwolfach Seminar: Analysis of Compressible Navier Stokes Equations and Related Topics

Organisers: Eduard Feireisl, Prague, David Gérard-Varet, Paris, Rupert Klein, Berlin, Antonin Novotny, Toulon

Programme: Main goal of the proposed Seminar is to present and discuss the state of the art of the mathematical theory of complete fluid systems, with the emphasis put on the underlying thermodynamical principles. Highlighting the new emerging trends we aim to inspire young researchers in their future activities. The Seminar activities will be initiated by a series of tutorial courses focused on the following topics:

• Fundamental problems of well-posedness and stability of the systems of partial differential equations arising in the modeling of compressible, viscous, and/or heat conducting fluids. Several concepts of weak solutions, admissibility criteria, weak formulation based on the laws of thermodynamics.

• Multiscale analysis of complete fluid systems, dependence on the shape of physical domains, the role of boundary conditions.

• Asymptotic behavior of solutions, their qualitative properties, singular limits

Modélisation de Fluides Compressibles pour la Terre

Mathematical modelling of compressible fluids

Mathematical modeling of compressible fluids real world problems requires models of complete fluid systems, where, in particular, the changes of density and of temperature are taken into account. These models have their origin in physical conservation laws of thermodynamics that are mathematically expressed in terms of (weak, strong, dissipative, measure-valued) solutions to the complete Navier-Stokes-Fourier system. In particular, they serve a basis for the modeling of the geophysical flows.

The main goal of the present project is to contribute to the development of a general mathematical theory of complete fluid systems based on the concept of weak solutions and formulated in accordance with the basic laws of thermodynamics, and to exploit this theory in various applications. The principal originality of this investigation is the use of the formulation of weak solutions of the *balance of entropy wit**h the entropy production rate beeing a Radon measure*. The crucial quantity in this analysis is a thermodynamic potential called *Helmholtz free function* and the associated *relative entropy functional* that allows to evaluate the *"distance" between a weak solution and any arbitray state of the fluid*. The principal tool is the *relative entropy inequality *governing the evolution of the relative entropy functional.

Within the ModTerCom project we shall develop these ideas in three directions described hereafter.

Stability and convergence of approximations and numerical schemes

Relative entropy inequality provides in a natural way an efficient tool to investigate the stability of strong solutions in the class of weak solutions with respect to the small perturbations of (arbitrary) large initial data. The class of weak solutions is sufficiently robust to incorporate the driving forces of very low regularity with respect to time. Consequently, it seems to be convenient for the investigation of the stability of small stochastic perturbations of deterministic driving forces. It also may be used as a measure of the "distance" of an approximate solution (given, in particular, by a numerical scheme) to an exact regular solution in order to obtain an error estimate.

The main objective of the project in the area of numerical analysis is to establish stable and convergent numerical methods by constructing a numerical analogy to the existence theory for the weak solutions. The final aim of these studies will be to extract from the relative entropy inequality all possible information about the numerical errors, rate of convergence and stability of the investigated numerical scheme. It may also serve as a selecting tool of "good" and efficient numerical schemes.

Multiscale analysis and model reduction

A specific aspect of the compressible fluid thermodynamics is the scale interaction. By scaling, meaning by choosing appropriately the reference units, the parameters determining the behavior of the system become explicit. Asymptotic analysis provides an useful tool in the situations when certain of these parameters called characteristic numbers vanish or become infinite.

We shall concentrate on the phenomena closely related to the geophysical and environmental fows. We name here some of the limits that have been studied in this situation: 1) passage from compressible to incompressible fluid flows (low Mach number limit); 2) passage from isotropic to anelastic fows (low Froude number limit); 3) passage from rotating 3D fows to planar fows (low Rossby number limit); 4) passage from viscous to inviscid fows (high Reynolds number limit).

Physically realistic regimes are however usually characterized by a simultaneous action of several parameters entering into the game and tending to extreme values in a particular mutual rate. The main idea in this part of the proposal is to use the relative entropy function for the evaluation of the "distance" between the weak solution of the complete Navier-Stokes system and a strong solution of the expected target equations describing the reduced model

Extreme value theory

We shall perform the analysis in two directions: 1) By studying global observables such as temperature, velocity, density and/or mean energy, via extreme value theory, the aim is to provide a detailed description of the nature of the observed extreme fuctuations. By associating a certain extreme value law to a particular fow regime it would be possible to recognize the transitions to diferent kind of states. This should help in identifying global stability thresholds that are nowadays only approximatively recognized. 2) Another question to address is how small and meso-scales structures affect the global statistical properties of the system. In particular we aim to and scaling laws for the parameters of extreme value distributions, and understand the different role between the region of quasi-laminar fows and the more turbulent regions.