Communications in Mathematical Sciences

Volume 12 (2014)

Number 8

Vortex patch problem for stratified Euler equations

Pages: 1541 – 1563

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n8.a8

Authors

Taoufik Hmidi (IRMAR, Université de Rennes 1, Campus de Beaulieu, Rennes, France)

Mohamed Zerguine (Département de Mathématiques, Faculté des Sciences, Université Hadj Lakhdar Batna, Batna, Algeria)

Abstract

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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