Well-posedness for Euler 2D in non-smooth domains

-
Christophe Lacave , Jussieu University
Fine Hall 314

The well-posedness of the Euler system has been of course the matter of many works, but a common point in all the previous studies is that the boundary is at least $C^{1,1}$. In a first part, we will establish the existence of global weak solutions of the 2D incompressible Euler equations for a large class of non-smooth open sets. For clarity, we will present the full details in the case where the domain is bounded and simply connected, and we will show the additional difficulties when we retrieve a finite number of obstacles (connected compact sets with positive capacity). In a second part, we will prove the uniqueness if the open set is a simply connected domain, where the boundary has a finite number of corners.  Although the velocity blows up near corners (for angles greater than $\pi$), we will get a similar theorem to the Yudovich's result, in the case of an initial vorticity with definite sign, bounded and compactly supported. The key point for the uniqueness part is to prove by a Liapounov energy that the vorticity never meets the boundary. Finally, we will present a uniqueness result without sign condition in the case of small corners.  The existence part is a work in collaboration with David Gerard-Varet, whereas the uniqueness part for small corners is in collaboration with Evelyne Miot and Chao Wang.