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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 $${C^{1+\varepsilon}}$$ C 1 + ε with $${0 < \varepsilon < 1}$$ 0 < ε < 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 $${\omega(t)={\bf1}_{\varOmega_t}+\widetilde\rho(t)}$$ ω ( t ) = 1 Ω t + ρ ~ ( t ) , where $${\varOmega_t=\varPsi(t,\varOmega_0)}$$ Ω t = Ψ ( t , Ω 0 ) keeps its initial regularity, with $${\varPsi(t,\cdot)}$$ Ψ ( t , · ) being the associated flow, and $${\widetilde\rho}$$ ρ ~ is a smooth function related to the smoothing effects of density.
Journal of Evolution Equations – Springer Journals
Published: Sep 1, 2015
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