Journal of Mathematical Fluid Mechanics

, Volume 17, Issue 1, pp 183–198 | Cite as

On Energy Inequality for the Problem on the Evolution of Two Fluids of Different Types Without Surface Tension

  • Irina Vlad. DenisovaEmail author


The paper deals with the motion of two immiscible viscous fluids in a container, one of the fluids being compressible while another one being incompressible. The interface between the fluids is an unknown closed surface where surface tension is neglected. We assume the compressible fluid to be barotropic, the pressure being given by an arbitrary smooth increasing function. This problem is considered in anisotropic Sobolev–Slobodetskiǐ spaces. We show that the L 2-norms of the velocity and deviation of compressible fluid density from the mean value decay exponentially with respect to time. The proof is based on a local existence theorem (Denisova, Interfaces Free Bound 2:283–312, 2000) and on the idea of constructing a function of generalized energy, proposed by Padula (J Math Fluid Mech 1:62–77, 1999). In addition, we eliminate the restrictions for the viscosities which appeared in Denisova (Interfaces Free Bound 2:283–312, 2000).

Mathematics Subject Classification

Primary 35Q30 76D05 Secondary 35R35 


Two-phase problem viscous compressible and incompressible fluids interface problem Navier–Stokes system Lagrangian coordinates energy inequality Sobolev–Slobodetskiǐ spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abels H.: On general solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound 9(1), 31–65 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bresch D., Perrin C., Zatorska E.: Singular limit of a Navier-Stokes system leading to a free/congested zones two-phase model. C. R. Math. Acad. Sci. Paris 352(9), 685–690 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Denisova I.V.: Evolution of compressible and incompressible fluids separated by a closed interface. Interfaces Free Bound 2(3), 283–312 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Denisova, I.V.: Evolution of closed interface between two liquids of different types. In: Proceedings of the 3ecm in Barcelona, July 2000, Birkhauser Verlag Basel. Prog. Math. 202, 263–272 (2001)Google Scholar
  5. 5.
    Denisova I.V.: Global L 2-solvability of a problem governing two-phase fluid motion without surface tension. Portugal. Math. 71(Fasc. 1), 1–24 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Denisova, I.V.: Problem of the motion of two compressible fluids separated by a closed free interface. Zap. Nauchn. Semin. Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 243, 61–86 (1997) [English transl. in J. Math. Sci. 99(1), 837–853 (2000)]Google Scholar
  7. 7.
    Denisova, I.V.: Solvability in weighted Hölder spaces for a problem governing the evolution of two compressible fluids. Zap. Nauchn. Semin. Petersburg. Otdel. Mat. Inst. Steklov. (POMI), 295, 57–89 (2003) [English transl. in J. Math. Sci. 127(2) (2005)]Google Scholar
  8. 8.
    Denisova, I.V.: A priori estimates of the solution of a linear time-dependent problem connected with the motion of a drop in a fluid medium. Trudy Mat. Inst. Steklov 188, 3–21 (1990) [English Proc. Steklov Intit. Math. no. 3, 1–24 (1991)]Google Scholar
  9. 9.
    Denisova I.V.: Problem of the motion of two viscous incompressible fluids separated by a closed free interface. Acta Appl. Math. 37, 31–40 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Denisova, I.V.: Global solvability of a problem on two fluid motion without surface tension. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 348, 19–39 (2007) [English transl. in J. Math. Sci. 152(5), 625–637 (2008)]Google Scholar
  11. 11.
    Denisova, I.V., Solonnikov, V.A.: Global solvability of a problem governing the motion of two incompressible capillary fluids. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 397, 20–52 (2011) [English transl. in J. Math. Sci. 185(5), 668-686 (2012)]Google Scholar
  12. 12.
    Köhne M., Prüss Ja., Wilke M.: Qualitative behaviour of solutions for the two-phase Navier- Stokes equations with surface tension. Math. Ann. 356(2), 737–792 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kubo, T., Shibata, Y., Soga, K.: On the R-boundedness for the two phase problem: compressible–incompressible model problem [to appear in Bound. Value Probl. (2014)]Google Scholar
  14. 14.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and quasilinear equations of parabolic type, Nauka, Moscow (1964) [English transl. in Am. Math. Soc., Providence, RI, 1968]Google Scholar
  15. 15.
    Ladyzhenskaya, O.A., Solonnikov, V.A.: Some problems of vector analysis and generalized statements of boundary value problems for Navier–Stokes equations. Zapiski nauchn. semin. LOMI 59, 81–116 (1976) [English transl. in J. Soviet Math. 10(2) (1978)]Google Scholar
  16. 16.
    Padula M.: On the exponential stability of the rest state of a viscous compressible fluid. J. Math. Fluid Mech. 1, 62–77 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  17. 17.
    Padula M.: Asymptotic Stability of Steady Compressible Fluids. Lect. Notes. Mat., vol. 2024, pp. 173. Springer, Berlin (2011)Google Scholar
  18. 18.
    Prüss J., Simonett G.: On the two-phase Navier–Stokes equations with surface tension. Interfaces Free Bound 12(3), 311–345 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Shibata Y., Shimizu S.: Maximal L p L q regularity for the two-phase Stokes equations; Model problems. J. Differ. Equ. 251, 373–419 (2011)CrossRefADSzbMATHMathSciNetGoogle Scholar
  20. 20.
    Solonnikov, V.A.: Estimates of the solution of non-stationary Navier–Stokes system. Zapiski nauchn. semin. LOMI 38, 153–231 (1973) [English transl. in J. Soviet Math. 8(4), 467–529 (1977)]Google Scholar
  21. 21.
    Solonnikov, V.A.: Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. Algebra i Analiz 3(1), 222–257 (1991) [English St.Petersburg Math.J. 3(1), 189–220 (1992)]Google Scholar
  22. 22.
    Solonnikov, V.A.: On non-stationary motion of an isolated mass of a viscous incompressible fluid. Isvestia Acad. Sci. USSR 51(5), 1065–1087 (1987) [English transl. in Math. USSR-Izv. 31(2), 381–405 (1988)]Google Scholar
  23. 23.
    Solonnikov, V.A., Tani, A.: Free boundary problem for a viscous compressible flow with surface tension. In: Constantin Carathéodory: An International Tribute, pp. 1270–1303. World Scientific (1991)Google Scholar
  24. 24.
    Solonnikov, V.A., Tani, A.: Evolution free boundary problem for equation of motion of viscous compressible barotropic liquid. In: Proceedings Oberwolfach 1991. Lect. Notes in Math. vol. 1530, pp. 30–55Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering, Russian Academy of SciencesSt. PetersburgRussia

Personalised recommendations