Analysis Seminar

Well-Posedness for Euler 2D in Non-Smooth Domains

Speaker: Christophe Lacave, Université Paris Diderot (Paris 7)

Location: Warren Weaver Hall 1302

Date: Thursday, November 17, 2011, 11 a.m.


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. These open sets are the complements (in a simply connected domain) of a finite number of connected compact sets with positive capacity. Existence of weak solutions with \(L^p\) vorticity is deduced from an approximation argument, that relates to the so-called \(\gamma\)-convergence of domains. In a second part, we will prove the uniqueness if the open set is the interior or the exterior of a simply connected domain, where the boundary has a finite number of corners. Although the velocity blows up near these corners, 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. The existence part is a work in collaboration with David Gerard-Varet.