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The regular vortex patch problem for stratified Euler equations with critical fractional dissipation

The regular vortex patch problem for stratified Euler equations with critical fractional dissipation 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

The regular vortex patch problem for stratified Euler equations with critical fractional dissipation

Journal of Evolution Equations , Volume 15 (3) – Sep 1, 2015

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References (28)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-015-0277-3
Publisher site
See Article on Publisher Site

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 $${C^{1+\varepsilon}}$$ 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

Journal of Evolution EquationsSpringer Journals

Published: Sep 1, 2015

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