On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

Takayuki  Kubo; Yoshihiro Shibata; Kohei  Soga

doi:10.3934/dcds.2016.36.3741

Article Contents

2016, Volume 36, Issue 7: 3741-3774. Doi: 10.3934/dcds.2016.36.3741 open access

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On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

1.
Division of Mathematics, University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8571
2.
Department of Mathematics and Research Institute of Science and Engineering, JST CREST, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555
3.
Department of Mathematics, Faculty of Science and Technology, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama, 223-8522

Received: April 30, 2015

Revised: November 30, 2015

Published: May 2017

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Abstract

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-bounded solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.

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Mathematics Subject Classification: Primary: 35K35; Secondary: 76N10.

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References

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