# 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

, Volume 73 (3) – Jun 1, 2016
22 pages

/lp/springer_journal/on-the-existence-for-the-free-interface-2d-euler-equation-with-a-muE7I0CGcK
Publisher
Springer US
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

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