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

show all references

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

[1]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[2]

Bingyan Liu, Xiongbing Ye, Xianzhou Dong, Lei Ni. Branching improved Deep Q Networks for solving pursuit-evasion strategy solution of spacecraft. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021016

[3]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[4]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[5]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[6]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[7]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[8]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[9]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[10]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[11]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[12]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[13]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

[14]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[15]

Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49

[16]

Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

[17]

Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311

[18]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[19]

El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in $ L^1 $ revisited. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 455-464. doi: 10.3934/dcdss.2020355

[20]

Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]