Abstract
We prove the existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions–Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier–Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solutions via an unconventional approximation using the Lagrangian formulation, truncations, and a stability result of trajectories for rough velocity fields.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Amosov, A.A., Zlotnik, A.A.: On the error of quasi-averaging of the equations of motion of a viscous barotoropic medium with rapidly oscillating data. Comput. Maths Phys. 36(10), 1415–1428 (1996)
Ben Belgacem, F., Jabin, P.-E.: Compactness for nonlinear continuity equations. J. Funct. Anal. 264(1), 139–168 (2013)
Bresch, D., Desjardins, B., Ghidaglia, J.-M., Grenier, E.: On Global weak Solutions to a generic two-fluid model. Arch. Ration. Mech. Anal. 196(2), 599–629 (2009)
Bresch, D., Desjardins, B., Ghidaglia, J.-M., Grenier, E., Hilliairet, M.: In: Giga, Y., Novotný, A. (eds.), pp. 1–52(2017)
Bresch, D., Hillairet, M.: Note on the derivation of multicomponent flow systems. Proc. AMS 143, 3429–3443 (2015)
Bresch, D., Hillairet, M.: A compressible multifluid system with new physical relaxation terms. To appear in Ann. Ecole Normale, Sup (2017)
Bresch, D., Huang, X.: A multi-fluid compressible system as the limit of weak solutions of the isentropic compressible Navier-Stokes equations. Arch. Ration. Mech. Anal. 201(2), 647–680 (2011)
Bresch, D., Huang, X., Li, J.: A global weak solution to a one-dimensional non-conservative viscous compressible two-phase system. Commun. Math. Phys. 309(3), 737–755 (2012)
Bresch, D., Jabin, P.-E.: Global existence of weak solutions for compresssible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor. Ann. Math. 188(2), 577–684 (2018)
Bresch, D., Jabin, P.-E.: Global weak solutions of PDEs for compressible media: a compactness criterion to cover new physical situations. Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics, Springer INdAM 17, 33–54 (2017)
Bresch, D., Jabin, P.-E.: Quantitative regularity estimates for advective equation with anelastic degenerate constraint. Proc, ICM Rio de Janeiro (2018)
Colombo, M., Crippa, G., Spirito, S.: Renormalized solutions to the continuity equation with an integrable damping term. Calc. Var. PDEs 54(2), 1831–1845 (2015)
Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna-Lions flows. J. Für Die Reine Und Angewandte Mathematik 616, 15–46 (2008)
Degond, P., Minakowski, P., Zatorska, E.: Transport of congestion in two-phase compressible/incompressible flows with. Nonlinear Anal. Real World Appl. 42, 485–510 (2018)
Degond, P., Minakowski, P., Navoret, L., Zatorska, E.: Finite Volume approximations of the Euler system with variable congestion. Comput. Fluids 169(30), 23–39 (2018)
Desvillettes, L.: Some Aspects of the Modeling at Different Scales of Multiphase Flows. Comput. Methods Appl. Mech. Eng. 199, 1265–1267 (2010)
Dias, F., Dutykh, D., Ghidaglia, J.-M.: A two-fluid model for violent aerated flows. Comput. Fluids 39(2), 283–293 (2010)
DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)
Drew, D., Passman, S.L.: Theory of multicomponent fluids. Applied Math Sciences, 1999
Evje, S.: An integrative multiphase model for cancer cell migration under influence of physical cues from the tumor microenvrionment. Chem Engineering Science, 204–259, 2017
Evje, S., Karlsen, K.H.: Global existence of weak solutions for a viscous two-phase model. J. Differ. Equ. 245(9), 2660–2703 (2008)
Evje, S., Wen, H.: Analysis of a compressible two-fluid stokes system with constant viscosity. J. Math. Fluid Mech. 17(3), 423–436 (2015)
Evje, S., Wen, H.: Stability of a compressible two-fluid hyperbolic-elliptic system arising in fluid mechanics. Nonlinear Anal. Real World Appl. 31, 610–629 (2016)
Evje, S., Wen, H., Yao, L.: Global solutions to a one-dimensional non-conservative two-phase model. Discrete Contin. Dyn. Syst. 36(4), 1927–1955 (2016)
Feireisl, E.: On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable. Comment. Math. Univ. Carolin 42(1), 83–98 (2001)
Feireisl, E., Klein, R., Novotný, A., Zatorska, E.: On singular limits arising in the scale analysis of stratified fluid flows. Math. Models Methods Appl. Sci. 26(3), 419–443 (2016)
Hao, C.C., Li, H.L.: Well-posedness for a multidimensional viscous liquid-gas flow model. SIAM J. Math. Anal. 44(3), 1304–1332 (2012)
Ishii, M., Hibiki, T.: Thermo-fluid dynamics of two-phase flow. Springer (2006)
Jabin, P.-E.: Differential Equations with singular fields. Journal de Mathématiques Pures et Appliquées 94(6), 597–621 (2010)
Kozono, H., Taniuchi, Y.: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Commun. Math. Phys. 214, 191–200 (2000)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol 1: Incompressible Models. New York: Oxford University Press, 1996
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol 2: Compressible Models. New York: Oxford University Press, 1998
Ogawa, T.: Sharp Sobolev Inequality of Logarithmic Type and the Limiting Regularity Condition to the Harmonic Heat Flow. SIAM J. Math. Anal. 34(6), 1318–1330 (2003)
Maltese, D., Michalek, M., Mucha, P.B., Novotny, A., Pokorny, M., Zatorska, E.: Existence of weak solutions for compressible Navier-Stokes equations with entropy transport. J. Differ. Equ. 261(8), 4448–4485 (2016)
Michálek, M.: Stability result for Navier-Stokes equations with entropy transport. J. Math. Fluid Mech. 17(2), 279–285 (2015)
Mucha, P.B.: The Cauchy problem for the compressible Navier-Stokes equations in the Lp-framework. Nonlinear Anal. 52(4), 1379–1392 (2003)
Mucha, P.B., Zajaczkowski, W.: On a \(L_p\)-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions. J. Differ. Equ. 186(2), 377–393 (2002)
Mucha, P.B., Zajaczkowski, W.: Global existence of solutions of the Dirichlet problem for the compressible Navier-Stokes equations. ZAMM Z. Angew. Math. Mech. 84(6), 417–424 (2004)
Serre, D.: Asymptotics of homogeneous oscillations in a compressible viscous fluid. Bull. Soc. Bras. Mat 32, 535–442 (2001)
Stein, E.M.: Maximal functions. I. Spherical means. Proc. Nat. Acad. Sci. USA 73(7), 2174–2175 (1976)
Vauchelet, N., Zatorska, E.: Incompressible limit of the Navier-Stokes model with a growth term. Nonlinear Anal. 163, 34–59 (2017)
Vasseur, A., Wen, H., Yu, C.: Global weak solution to the viscous two-phase model with finite energy. Preprint, arXiv:1704.07354
Acknowledgements
The author D.B. is partly supported by the ANR- 13-BS01- 0003-01 project DYFICOLTI, by the ANR-16-CE06-0011-02 FRAISE and by the project TelluS-INSMI-MI (INSU) CNRS. The authors P.B.M. and E.Z. have been partly supported by National Science Centre grant 2014/14/M/ST1/00108 (Harmonia). The authors want to thank P.E. Jabin and the anonymous referee for their valuable remarks about the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T.-P. Liu
Rights and permissions
About this article
Cite this article
Bresch, D., Mucha, P.B. & Zatorska, E. Finite-Energy Solutions for Compressible Two-Fluid Stokes System. Arch Rational Mech Anal 232, 987–1029 (2019). https://doi.org/10.1007/s00205-018-01337-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-018-01337-6