# American Institute of Mathematical Sciences

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The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics
September  2010, 3(3): 429-449. doi: 10.3934/dcdss.2010.3.429

## A simple proof of well-posedness for the free-surface incompressible Euler equations

 1 Maxwell Institute for Mathematical Sciences and department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom 2 Department of Mathematics, University of California, Davis, CA 95616

Received  November 2009 Revised  January 2010 Published  May 2010

The purpose of this this paper is to present a new simple proof for the construction of unique solutions to the moving free-boundary incompressible 3-D Euler equations in vacuum. Our method relies on the Lagrangian representation of the fluid, and the anisotropic smoothing operation that we call horizontal convolution-by-layers. The method is general and can be applied to a number of other moving free-boundary problems.
Citation: Daniel Coutand, Steve Shkoller. A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 429-449. doi: 10.3934/dcdss.2010.3.429
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