Advertisement

About Compressible Viscous Fluid Flow in a 2-dimensional Exterior Domain

  • Yuko EnomotoEmail author
  • Yoshihiro Shibata
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)

Abstract

We report our results [5, 6, 7] concerning a global in time unique existence theorem of strong solutions to the equation describing the motion of compressible viscous fluid flow in a 2-dimensional exterior domain for small initial data and some decay properties of the analytic semigroup associated with Stokes operator of compressible viscous fluid flow in a 2-dimensional exterior domain.Our results are an extension of the works due to Matsumura and Nishida [13] and Kobayashi and Shibata [10] in a 3-dimensional exterior domain to the 2-dimensional case.W e also discuss some analytic semigroup approach to the compressible viscous fluid flow in a bounded domain, which was first investigated by G.S trömer [20, 21, 22].

Keywords

2-dimensional exterior domain global in time unique existence theorem local energy decay Lp-Lq decay estimate 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    W. Borchers and W. Varnhorn, On the boundedness of the Stokes semigroup in two-dimensional exterior domains, Math. Z., 213 (1993), 275-299.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    W. Dan and Y. Shibata, On the Lq -Lr estimates of the Stokes semigroup in a two-dimensional exterior domain, J. Math. Soc. Japan, 51 (1999), no. 1, 181-207.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domain, Math. Z., 209 (1992), 115-130.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    K. Deckelnick, L2-decay for the compressible Navier-Stokes equations in unbounded domains, Comm. Partial Differential Equations, 18 (1993), 1445-1476.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Y. Enomoto and Y. Shibata, On some decay properties of Stokes semigroup of compressible viscous fluid flow in a two-dimensional exterior domain, preprint.Google Scholar
  6. 6.
    Y. Enomoto, Y. Shibata and M. Suzuki, On the maximal Lp-Lq regularity of the Stokes semigroup of compressible viscous fluid flow and its application to a nonlinear problem, preprint.Google Scholar
  7. 7.
    Y. Enomoto, Y. Shibata and M. Suzuki, On the global in time unique existence theorem for the compressible viscous fluid flow in 2-dimensional exterior domains, preprint.Google Scholar
  8. 8.
    R. Kleinman and B. Vainberg, Full low-frequency asymptotic expansion for second-order elliptic equations in two dimensions, Math. Meth. Appl. Sci., 17 (1994), 989-1004.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    T. Kobayashi, On a local energy decay of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3, Tsukuba J. Math., 21 (1997), No. 3, 629-670.MathSciNetzbMATHGoogle Scholar
  10. 10.
    T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3, Commun. Math. Phys., 200 (1999), 621-659.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    P. Maremonti and V.A. Solonnikov, On nonstationary Stokes problem in exterior domain, Ann. Scuola Norm. Sup. Pisa CI. Sci., 24 (1997) no. 4, 395-449.MathSciNetzbMATHGoogle Scholar
  12. 12.
    A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 339-418.MathSciNetCrossRefGoogle Scholar
  15. 15.
    R.T. Seeley, Integral equations depending analytically on a parameter, Indag. Math., 24 (1964), 434-443.MathSciNetGoogle Scholar
  16. 16.
    Y. Shibata and S. Shimizu, On a free boundary problem for the Navier-Stokes equations, Diff.Int. Eqns., 20 (2007), 241-276.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Y. Shibata and S. Shimizu, On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, J. reine angew. Math., 615 (2008), 157-209 (CRELLE).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Y. Shibata and S. Shimizu, On the maximal Lp-Lq regularity of the Stokes problem with first-order boundary condition: Model Problem, to appear in J. Math. Soc. Japan.Google Scholar
  19. 19.
    Y. Shibata and K. Tanaka, On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion, Math. Methods Appl. Sci., 27 (2004) no. 13, 1579-1606.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    G. Ströhmer, About the resolvent of an operator from fluid dynamics, Math. Z., 194 (1987), 183-191.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    G. Ströhmer, About a certain class of parabolic-hyperbolic systems of differential equations, Analysis, 9 (1989), 1-39.MathSciNetzbMATHGoogle Scholar
  22. 22.
    G. Ströhmer, About compressible viscous fluid flow in a bounded region, Pacific J. Math., 143 (1990) no. 2, 359-375.MathSciNetzbMATHGoogle Scholar
  23. 23.
    A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. RIMS Kyoto Univ., 13 (1977), 193-253.zbMATHCrossRefGoogle Scholar
  24. 24.
    B. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, in Russian, Moscow Univ. Press, 1982; Gordon and Breach Publishers, New York, London, Paris, Montreux, Tokyo, 1989; English translation.Google Scholar
  25. 25.
    L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann., 319 (2001), 735-758.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Saitama-shiJapan
  2. 2.TokyoJapan

Personalised recommendations