Abstract
This article focuses on studying the problem of a regular vortex patch for the two-dimensional stratified Euler system with critical fractional dissipation. We exhibit that if the initial density is smooth function and the boundary of the initial vortex patch belongs to the space \({C^{1+\varepsilon}}\) with \({0 < \varepsilon < 1}\), then the corresponding velocity is a Lipschitz function globally in time. We provide also that the advected vorticity can be decomposed into two parts, namely we have \({\omega(t)={\bf1}_{\varOmega_t}+\widetilde\rho(t)}\), where \({\varOmega_t=\varPsi(t,\varOmega_0)}\) keeps its initial regularity, with \({\varPsi(t,\cdot)}\) being the associated flow, and \({\widetilde\rho}\) is a smooth function related to the smoothing effects of density.
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Zerguine, M. The regular vortex patch problem for stratified Euler equations with critical fractional dissipation. J. Evol. Equ. 15, 667–698 (2015). https://doi.org/10.1007/s00028-015-0277-3
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DOI: https://doi.org/10.1007/s00028-015-0277-3