# American Institute of Mathematical Sciences

July  2016, 36(7): 3741-3774. doi: 10.3934/dcds.2016.36.3741

## 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, Japan 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, Japan

Received  May 2015 Revised  December 2015 Published  March 2016

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.
Citation: Takayuki Kubo, Yoshihiro Shibata, Kohei Soga. On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3741-3774. doi: 10.3934/dcds.2016.36.3741
##### References:
 [1] I. V. Denisova, Evolution of compressible and imcompressible fluids separated by a closed interface,, Interface Free Bound., 2 (2000), 283.  doi: 10.4171/IFB/21.  Google Scholar [2] I. V. Denisova and V. A. Solonnikov, Classical solvability of a problem on the motion of an isolated mass of a compressible liquid,, St. Petersburg Math. J., 14 (2003), 53.   Google Scholar [3] I. V. Denisova and V. A. Solonnikov, Classical solvability of a model problem in a half-space, related to the motion of an isolated mass of a compressible fluid,, J. Math. Sci., 115 (2003), 2753.  doi: 10.1023/A:1023365718404.  Google Scholar [4] R. Denk, M. Hieber and J. Prüß, $\mathcalR$-boundedness, Fourier multiplier and problems of elliptic and parabolic type,, Memories of AMS., 166 (2003).  doi: 10.1090/memo/0788.  Google Scholar [5] Y. Enomoto, L. v. Below and Y. Shibata, On some free boundary problem for a compressible barotopic viscous fluid flow,, Ann Univ Ferrara, 60 (2014), 55.  doi: 10.1007/s11565-013-0194-8.  Google Scholar [6] Y. Enomoto and Y. Shibata, On the $\mathcalR$-sectoriality and its application to some mathematical study of the viscous compressible fluids,, Funkcial. Ekvac., 56 (2013), 441.  doi: 10.1619/fesi.56.441.  Google Scholar [7] D. Götz and Y. Shibata, On the $\mathcalR$-boundedness of the solution operators in the study of the compressible viscous fluid with free boundary condition,, Asymptotic Analysis, 90 (2014), 207.  doi: 10.3233/ASY-141238.  Google Scholar [8] T. Kubo, Y. Shibata and K. Soga, On the $\mathcalR$-boundedness for the Two phase prolem: Compressible-incompressible model prolem,, Boundary Value Problems, 2014 (2014).  doi: 10.1186/s13661-014-0141-3.  Google Scholar [9] P. Scchi and A. Valli, A free boundary problem for compressible viscous fluid,, J. Reine Angew, 341 (1983), 1.  doi: 10.1515/crll.1983.341.1.  Google Scholar [10] Y. Shibata and S. Shimizu, On the $L_p$-$L_q$ maximal regularity of the Neumann problem for the Stokes equations in a bounded domain,, J.Reine Angew. Math., 615 (2008), 157.  doi: 10.1515/CRELLE.2008.013.  Google Scholar [11] Y. Shibata and K. Tanaka, On a resolvent problem for the linealized system from the dynamical system describing the compressible viscous fluid motion,, Math. Mech. Appl. Sci., 27 (2004), 1579.  doi: 10.1002/mma.518.  Google Scholar [12] V. A. Solonnikov and A. Tani, Free boundary problem for a viscous compressible flow with the surface tension,, Constantin Carathéodory: An International Tribute (Ih. M. Rassias, (1991), 1270.   Google Scholar [13] G. Ströhmer, About the resolvent of an operator from fluid dynamics,, Math. Z., 194 (1987), 183.  doi: 10.1007/BF01161967.  Google Scholar [14] A. Tani, On the free boundary value problem for compressible viscous fluid motion,, J. Math. Kyoto Univ., 21 (1981), 839.   Google Scholar [15] A. Tani, Two-phase free boundary problem for compressible viscous fluid motion,, J. Math. Kyoto Univ., 24 (1984), 243.   Google Scholar [16] L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity,, Math. Ann., 319 (2001), 735.  doi: 10.1007/PL00004457.  Google Scholar

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##### References:
 [1] I. V. Denisova, Evolution of compressible and imcompressible fluids separated by a closed interface,, Interface Free Bound., 2 (2000), 283.  doi: 10.4171/IFB/21.  Google Scholar [2] I. V. Denisova and V. A. Solonnikov, Classical solvability of a problem on the motion of an isolated mass of a compressible liquid,, St. Petersburg Math. J., 14 (2003), 53.   Google Scholar [3] I. V. Denisova and V. A. Solonnikov, Classical solvability of a model problem in a half-space, related to the motion of an isolated mass of a compressible fluid,, J. Math. Sci., 115 (2003), 2753.  doi: 10.1023/A:1023365718404.  Google Scholar [4] R. Denk, M. Hieber and J. Prüß, $\mathcalR$-boundedness, Fourier multiplier and problems of elliptic and parabolic type,, Memories of AMS., 166 (2003).  doi: 10.1090/memo/0788.  Google Scholar [5] Y. Enomoto, L. v. Below and Y. Shibata, On some free boundary problem for a compressible barotopic viscous fluid flow,, Ann Univ Ferrara, 60 (2014), 55.  doi: 10.1007/s11565-013-0194-8.  Google Scholar [6] Y. Enomoto and Y. Shibata, On the $\mathcalR$-sectoriality and its application to some mathematical study of the viscous compressible fluids,, Funkcial. Ekvac., 56 (2013), 441.  doi: 10.1619/fesi.56.441.  Google Scholar [7] D. Götz and Y. Shibata, On the $\mathcalR$-boundedness of the solution operators in the study of the compressible viscous fluid with free boundary condition,, Asymptotic Analysis, 90 (2014), 207.  doi: 10.3233/ASY-141238.  Google Scholar [8] T. Kubo, Y. Shibata and K. Soga, On the $\mathcalR$-boundedness for the Two phase prolem: Compressible-incompressible model prolem,, Boundary Value Problems, 2014 (2014).  doi: 10.1186/s13661-014-0141-3.  Google Scholar [9] P. Scchi and A. Valli, A free boundary problem for compressible viscous fluid,, J. Reine Angew, 341 (1983), 1.  doi: 10.1515/crll.1983.341.1.  Google Scholar [10] Y. Shibata and S. Shimizu, On the $L_p$-$L_q$ maximal regularity of the Neumann problem for the Stokes equations in a bounded domain,, J.Reine Angew. Math., 615 (2008), 157.  doi: 10.1515/CRELLE.2008.013.  Google Scholar [11] Y. Shibata and K. Tanaka, On a resolvent problem for the linealized system from the dynamical system describing the compressible viscous fluid motion,, Math. Mech. Appl. Sci., 27 (2004), 1579.  doi: 10.1002/mma.518.  Google Scholar [12] V. A. Solonnikov and A. Tani, Free boundary problem for a viscous compressible flow with the surface tension,, Constantin Carathéodory: An International Tribute (Ih. M. Rassias, (1991), 1270.   Google Scholar [13] G. Ströhmer, About the resolvent of an operator from fluid dynamics,, Math. Z., 194 (1987), 183.  doi: 10.1007/BF01161967.  Google Scholar [14] A. Tani, On the free boundary value problem for compressible viscous fluid motion,, J. Math. Kyoto Univ., 21 (1981), 839.   Google Scholar [15] A. Tani, Two-phase free boundary problem for compressible viscous fluid motion,, J. Math. Kyoto Univ., 24 (1984), 243.   Google Scholar [16] L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity,, Math. Ann., 319 (2001), 735.  doi: 10.1007/PL00004457.  Google Scholar
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