Vortex patch problem for stratified euler equations

We study in this paper the vortex patch problem for the stratified Euler equations in space dimension two. We generalize Chemin's result [J.Y. Chemin, Oxford University Press, 1998.] concerning the global persistence of the Ḧolderian regularity of the vortex patches. Roughly speaking, we prove that if the initial density is smooth and the initial vorticity takes the form ω0=1_Ω, with Ω a C^1+ε -bounded domain, then the velocity of the stratified Euler equations remains Lipschitz globally in time and the vorticity is split into two parts ω(t)=1_Ωt+ρ{ogonek}(t), where Ω_t denotes the image of Ω by the flow and has the same regularity of the domain Ω. The function ρ{ogonek} is a smooth function.