APDE seminar@BCAM: The non-homogeneous Euler equations below the Lipschitz threshold

Abstract

The incompressible Euler equations are well-known to be globally well-posed in the case of space dimension 2, both in the strong  solutions framework and in the Yudovich framework. No results of that  kind are known for the non-homogeneous (that is, density-dependent)  incompressible Euler system. 

In this talk, we show that both problems (i.e., global well-posedness  and theory à la Yudovich for the density-dependent case) can be  reduced to the study of a non-linear geometric quantity, which encodes  the regularity of the velocity field along the level lines of the  density. Such a geometric regularity places itself below the Lipschitz  threshold.