Important progress has been made on the construction of nonlinear Ekman layers. In collaboration with D. Gérard-Varet and Y. Maekawa, the PI has proved the existence and uniqueness of solutions of nonlinear boundary layer equations for rotating fluids, in general environments.
The PI has also obtained a breakthrough result with N. Masmoudi on boundary layer separation. They obtained the first mathematical proof of separation for the stationary Prandtl equation, and showed the relevance of the “Goldstein singularity”. The corresponding paper has been published in Publications mathématiques de l'IHES.
The techniques developed by the PI and N. Masmoudi could likely be extended to western boundary layers in oceanography, and therefore give a mathematical description of separation of the Gulf Stream.
On the other hand, the PI and M. Paddick studied the stabilizing effect of rotation on western boundary layers and gave a quantitative condition on the profile of the coastline that prevents recirculation.
Concerning boundary layer denegeracy, the PI and L. Saint-Raymond have solved the problem of geostrophic degeneracy. J. Rax studied several degenerate boundary layer problems: the equatorial Ekman layer, boundary layers in MHD, and recirculating solutions of the stationary Burgers equation with transverse viscosity.
A team composed of the PI, H. Dietert, D. Gérard-Varet and F. Marbach has also studied alternate boundary layer models, which were claimed to have a better behavior than the Prandtl system. However, the afore mentioned team proved that the behaviour of the time-dependent versions of these models is actually worse: there are less stable profiles, and the instabilities present in the models are stronger than the ones for Prandtl.
G. Lopez-Ruiz proved the well-posedness of the boundary layer equations for western boundary currents in the vicinity of rough shores.
Other achievements include:
- Criteria for the convergence of weak solutions of the 2D incompressible Navier-Stokes equations with Navier slip boundary conditions (M. Paddick, Y. Maekawa)
- Quantitative description of critical reflection phenomena for internal waves (R. Bianchini, L. Saint-Raymond, PI)
- Stability properties of traveling fronts within a soft congestion model (C. Perrin, PI)
- 1d compressible Euler equations with a singular pressure law (R. Bianchini, C. Perrin)
- Linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids (R. Bianchini, M. Coti-Zelati, M. Dolce)
- Global stability of a class of semilinear waves in 2+1 dimension under a null condition and global stability of a wave and Klein-Gordon system with mixed coupling (S. Dong)
- Correction to Einstein's formula for the effective viscosity in a dilute suspension of particles (A. Mecherbet, D. Gérard-Varet)
- A nalysis of a transport-Stokes equation describing sedimentation of a droplet in viscous fluid (A. Mecherbet)
