The project is organized around four important topics in fluid mechanics: free surfaces and interfaces, boundary layers, vortex dynamics and fluid-structure interactions. The mathematical and the physical-environmental motivations of the project are connected to the events of the 2013 Mathematics of Planet Earth program, which will also suggest new directions of research. The four topics are closely interconnected because they often coexist in the same physical situation, and because the mathematical tools (such as multiscale analysis, asymptotic expansions, or stability theory) that are needed to analyze them are quite similar. Our main directions of research will be:
We are mostly interested in situations which are singular, either because of the lack or smoothness (examples are wave breaking, the description of shorelines, and the influence of rough topographies in shallow water models), or because of the presence of small parameters (examples are continuous but sharp stratification in two fluid models, multiscale models that describe the energy spectrum in wave breaking, and compressible fluids with free surface at low Mach number). We expect improvements in the modelling and numerical simulations of these phenomena through the derivation of more accurate asymptotic models. We also plan to develop suitable mathematical tools in order to handle these singular situations rigorously.
We are interested both in the construction of boundary layers expansions and in the study of their stability properties. For the first aspect, we shall study the construction of boundary layers in degenerate situations, for example in the presence of rough boundaries or in situations where boundary layers of different sizes need to be connected (this is crucial to understand oceanic circulation). We shall also study the well-posedness of Prandtl-type equations that arise in oceanics models. For the second aspect, we plan to make progress in the understanding of instabilities in boundary layers, either in the classical inviscid limit of the incompressible Navier-Stokes equation with Dirichlet boundary condition by addressing the question of the destabilizing effect of viscosity, or in slightly regularized situations like some critical Navier conditions or the alphamodels equations.
We shall study both perfect and viscous incompressible fluids, using mainly the vorticity equation. Our interest lies in singular domains (flow around rough obstacles) or in singularly perturbed domains (flow around small obstacles). In the two-dimensional case, the question of understanding the longtime behaviour of perfect and viscous fluids will be also addressed. Another important direction of research will be the study of vortex filaments, the most challenging question being the rigorous understanding of the motion of vortex filaments in the vanishing viscosity limit (the expected asymptotic model is the binormal flow).
We first plan to get a better understanding of qualitative properties of the fluid-structure interactions on the most simple models (incompressible fluids with rigid bodies). Typical questions that will be addressed are the uniqueness of weak solutions in 2D for viscous fluids, or the smoothness of particles trajectories. Singular limits such as the vanishing viscosity limit, the small obstacle limit, or the mean field limit will be also studied. Finally we plan to make progress in the understanding of more complete models that take into account for example deformable solids or compressible fluids.