On the Existence for the Free Interface 2D Euler Equation with a Localized Vorticity Condition

On the Existence for the Free Interface 2D Euler Equation with a Localized Vorticity Condition We prove a local-in-time existence of solutions result for the two dimensional incompressible Euler equations with a moving boundary, with no surface tension, under the Rayleigh–Taylor stability condition. The main feature of the result is a local regularity assumption on the initial vorticity, namely $$H^{1.5+\delta }$$ H 1.5 + δ Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial fluid velocity in the $$H^{2+\delta }$$ H 2 + δ space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in $$H^{2+\delta }$$ H 2 + δ . The assumptions on the initial data constitute the minimal set of assumptions for the existence of solutions to the rotational flow problem to be established in 2D. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

On the Existence for the Free Interface 2D Euler Equation with a Localized Vorticity Condition

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-016-9346-4
Publisher site
See Article on Publisher Site

Abstract

We prove a local-in-time existence of solutions result for the two dimensional incompressible Euler equations with a moving boundary, with no surface tension, under the Rayleigh–Taylor stability condition. The main feature of the result is a local regularity assumption on the initial vorticity, namely $$H^{1.5+\delta }$$ H 1.5 + δ Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial fluid velocity in the $$H^{2+\delta }$$ H 2 + δ space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in $$H^{2+\delta }$$ H 2 + δ . The assumptions on the initial data constitute the minimal set of assumptions for the existence of solutions to the rotational flow problem to be established in 2D.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2016

References

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