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September  2019, 8(3): 503-542. doi: 10.3934/eect.2019025

A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid

Marcelo M. Disconzi 1,, and  Igor Kukavica 2,

1. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
2. 

Department of Mathematics, University of Southern California, Los Angeles, CA 91107, USA

* Corresponding author: marcelo.disconzi@vanderbilt.edu

Received  April 2018 Revised  January 2019 Published  September 2019 Early access  May 2019

Fund Project: The first author is partially supported by the NSF grant DMS-1812826, a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Discovery grant administered by Vanderbilt University. The second author is partially supported by the NSF grant DMS-1615239.

We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial velocity and initial density belong to H3, with an extra regularity condition on the moving boundary, thus lowering the regularity of the initial data. Our methods are direct and involve two key elements: the boundary regularity provided by the mean curvature and a new compressible Cauchy invariance.

Keywords: Compressible Euler, free-boundary, surface tension, liquid, regularity.
Mathematics Subject Classification: Primary: 35L45; Secondary: 35L60.
Citation: Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025
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