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 Hölderian regularity of the vortex patches. Roughly speaking, we prove that if the initial density is smooth and the initial vorticity takes the form ${\omega}_0 = 1_{\Omega}$ with $\Omega$ a $C^{1+\epsilon}$-bounded domain, then the velocity of the stratified Euler equations remains Lipschitz globally in time and the vorticity is split into two parts $\omega (t) = 1_{{\Omega}_t} + \tilde{\rho}(t)$, where ${\Omega}_t $ denotes the image of $\Omega$ by the flow and has the same regularity of the domain $\Omega$. The function $\tilde{\rho}$ is a smooth function.

Keywords

stratified system, vortex patches, para-differential calculus, time decay

2010 Mathematics Subject Classification

35B65, 35Q35, 76D03

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Published 14 May 2014