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

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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 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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