Fluid dynamics, boundary layers, vortices and interfaces – DYFICOLTI

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 Panet 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, stability theory) that are needed to analyze them are quite similar. Our main directions of research will be:

-Free surfaces and interfaces. We are mostly interested in situations which are singular, either because of the lack of 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.

-Boundary layers. We are interested both in the construction of boundary layers expansions and 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 the 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 alpha-models equations

-Vortex dynamics. We shall study both perfect and viscous incompressible fluids using mainly the vorticity equation. Our interest lies in singular domains (flow around rough obstacles for example) or in singularly pertubed domains (flow around small obstacles). In the two-dimensional case, the question of understanding the large time behaviour of perfect and viscous fluids will be also adressed. 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).

-Fluid-structure interactions. 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 and the study of the smoothness of particles trajectories. Some singular limits like vanishing viscosity limit, vanishing particles limit and 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.