INhomogeneous Flows : Asymptotic Models and Interfaces Evolution – INFAMIE

Our project aims at a better mathematical understanding of several models for the evolution of inhomogeneous flows. Through three main lines of research (see below), we will pursue a twofold final objective. First, we want to develop the current theory of regular solutions for several equations for the evolution of fluids, proposing a new approach and developing tools that are likely to be efficient in various areas of PDEs. Second, for a few selected concrete systems that describe flows in the earth environment or in astrophysics, we wish to use this general approach to extract as much information as possible concerning the qualitative behavior of the solutions.
Our first line of research is to provide a rigorous and quantitative justification of simplified models derived from the full Navier-Stokes system that are used in some applications. The usual and natural approach often consists in neglecting the terms that are expected to correspond to very small physical effects. Typical examples are given by incompressible models applied in meteorology although air is not incompressible. We plan to investigate the domain of validity of models that are e.g. used in the low Mach number limit asymptotic, in the diffusive limit for radiating flows, or in the semi-classical limit for fluids endowed with internal capillarity.

Our second line of research concerns the interface dynamics for mixtures of non-reacting flows. Following a recent work by two members of the project, we propose to use Lagrangian coordinates in order to study whether the `global model', where interfaces are not prescribed in advance, allows to get relevant qualitative informations on the evolution of mixtures. Both for numerical and theoretical purposes, the possibility to use the global model in this situation would be a great improvement,as one just has to solve a PDE in a fixed domain rather than a free boundary problem with complicated transmission conditions.

Our third line is to investigate the evolution of fluids in bounded or exterior domains and to compare their dynamics with those of the whole space case. This is a fundamental issue for, in most applications, the fluid domain has nonempty borders, on which (physical) boundary conditions have to be prescribed. The main difficulty we have to face is that this first means to develop new analytical tools so as to solve the fluid equations in domains, with the same accuracy as in the whole space.

The common setting between these three lines of research is the striving for a critical functional framework corresponding to the equation under consideration. Indeed, considering only critical norms or related quantities often yields the optimal results for the study of the Initial Value Problem, the derivation of blow-up (or continuation) criteria and the asymptotic properties of the solutions. Since the 80s, with the growing success of Fourier analysis methods, the critical regularity approach has led to substantial progress in the study of evolution equations for fluids. However, Fourier analysis methods collapse if the fluid domain is not the whole space or the torus. To extend the critical regularity approach to the domain case, we will first have to imagine a good substitute to Fourier analysis. This is a very challenging issue, which is obviously also relevant for other topics than equations for fluids.
It could be achieved by improving some endpoint parabolic type maximal regularity estimates (needed in our third objective), as well as extending them to the rough coefficients case (for the second objective). In this regard, encouraging preliminary results have been obtained recently by some members of the project.