Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

Chengchun  Hao

doi:10.3934/dcdsb.2015.20.2885

Article Contents

2015, Volume 20, Issue 9: 2885-2931. Doi: 10.3934/dcdsb.2015.20.2885

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Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

Chengchun Hao^1,

1.
Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190

Received: May 31, 2014

Revised: May 31, 2015

Published: October 2015

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Abstract

In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.

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Mathematics Subject Classification: Primary: 35R35, 35B45; Secondary: 35Q35.

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