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

Marcelo M. Disconzi; Igor Kukavica

doi:10.3934/eect.2019025

Article Contents

2019, Volume 8, Issue 3: 503-542. Doi: 10.3934/eect.2019025

This issue Previous Article A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments Next Article A comparison principle for Hamilton-Jacobi equation with moving in time boundary

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
^* Corresponding author: marcelo.disconzi@vanderbilt.edu

Received: March 31, 2018

Revised: December 31, 2018

Early access: May 2019

Published: August 2019

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L45; Secondary: 35L60.

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