Skip to main content
SpringerLink
Log in

Existence of Stationary Weak Solutions for the Isentropic and Isothermal Flows

  • Living reference work entry
  • First Online:
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

  • 201 Accesses

Abstract

The aim of this chapter is to study the existence of weak solutions to the multidimensional steady isentropic and isothermal compressible Navier-Stokes equations with large external forces. In the past decades, significant progress has been made on the existence of large weak solutions. In this chapter, a brief review of recent existence results on the existence of (renormalized) stationary weak solutions with large external forces will be presented. Different boundary value problems, such as the spatially periodic, slip and Dirichlet boundary value problems, will be investigated, and some related topics such as nonuniqueness, regularity, etc., will also be discussed. The ideas and developed techniques used in analysis will be presented and analyzed, and some open problems will be addressed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Similar content being viewed by others

References

  1. D.R. Adams, A note on Riesz potentials. Duke Math. J. 42, 765–778(1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory (Springer, Berlin/Heidelberg/New York, 1996)

    Book  MATH  Google Scholar 

  3. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying gerneral boundary conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MATH  Google Scholar 

  4. M.E. Bogovskii, Solution of the first boudary value problem for the equation of continuity of an incompressible medium. (Russian) Dokl. Akad. Nauk SSSR 248(5), 1037–1040 (1979)

    Google Scholar 

  5. J. Březina, A. Novotný, On weak solutions of steady Navier-Stokes equations for monoatomic gas. Comment. Math. Univ. Carolinae. 49(4), 611–632 (2008)

    MathSciNet  MATH  Google Scholar 

  6. F. Chiarenza, M. Frasca, Morrey spaces and Hardy – Littlewood maximal function. Rend. Mat. Appl. VII. Ser. 7(3–4), 273–279 (1987)

    MathSciNet  MATH  Google Scholar 

  7. H. Choe, B. Jin, Existence of solutions of stationary compressible Navier-Stokes equations with large force. J. Funct. Anal. 177, 54–88 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Coifman, P.L. Lions, Y. Meyer, S. Semmes, Compensated compactness and hardy spaces. J. Math. Pure Appl. 72, 247–86 (1993)

    MathSciNet  MATH  Google Scholar 

  9. G. Da Prato, Spazi \(\mathcal{L}(\varOmega,\delta )\) e loro proprietá. Ann. Math. Pura Appl. 69, 383–392 (1965)

    Article  Google Scholar 

  10. R.J. Diperna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Dou, F. Jiang, S. Jiang, Y. Yang, Existence of strong solutions to the steady Navier-Stokes equations for a compressible heat-conductive fluid with large forces. J. Math. Pures Appl. 103, 1163–1197 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Dutto, A. Novotný, Physically reasonable solutions to steady compressible Navier-Stokes equations in 2d exterior domains with nonzero velocity at infinity. J. Math. Fluid Mech. 3, 99–138 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable. Comment. Math. Univ. Carolinae. 42, 83–98 (2001)

    MathSciNet  MATH  Google Scholar 

  14. E. Feireisl, H. Petzeltová, On compactness of solutions to the Navier-Stokes equations of compressible flow. J. Differ. Equ. 163, 57–75 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Frehse, S. Goj, M. Steinhauer, L p-estimates for the Navier-Stokes equations for steady compressible flow. Manuscripta Math. 116, 265–275 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Frehse, M. Steinhauer, W. Weigant, On stationary solutions for 2-D viscous compressible isothermal Navier-Stokes equations. J. Math. Fluid Mech. 13, 55–63 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for viscous compressible isothermal Navier-Stokes equations in two-dimensions. Arch. Ration. Mech. Anal. 198, 1–12 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Frehse, M. Steinhauer, W. Weigant, The Dirichlet problem for steady viscous compressible flow in 3-D. J. Math. Pures Appl. 97, 85–97 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. G.P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations, vol. 1 (Springer, Berlin, 1994)

    MATH  Google Scholar 

  21. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn. (Springer, Berlin/Heidelberg/New York, 1983)

    Book  MATH  Google Scholar 

  22. Y. Guo, S. Jiang, C. Zhou, Steady viscous compressible channel flows. SIAM J. Math. Anal. 47(5), 3648–3670 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. J.G. Heywood, On uniqueness questions in the theory of viscous flow. Acta Math. 136, 61–102 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  24. D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Ration. Mech. Anal. 132, 1–14 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Jesslé, A. Novotný, Existence of renormalized weak solutions to the steady equations describing compressible fluids in barotropic regime. J. Math. Pures Appl. 99, 280–296 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Jiang, P. Zhang, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Commun. Math. Phys. 215, 559–581 (2001)

    Article  MATH  Google Scholar 

  27. S. Jiang, C. Zhou, Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 28, 485–498 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. C.H. Jun, J.R. Kweon, For the stationary compressible viscous Navier-Stokes equations with no-slip condition on a convex polygon. J. Differ. Equ. 250, 2440–2461 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. R.B. Kellogg, J.R. Kweon, Compressible Navier-Stokes equations in a bounded domain with inflow boundary condition. SIAM J. Math. Anal. 28, 94–108 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. R.B. Kellogg, J.R. Kweon, Regularity of solutions to the Navier-Stokes equations for compressible barotropic flows on a polygon. Arch. Ration. Mech. Anal. 163, 35–64 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. R.B. Kellogg, J.R. Kweon, Regularity of solutions to the Navier-Stokes system for compressible flows on a polygon. SIAM J. Math. Anal. 35, 1451–1485 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. J.R. Kweon, A jump discontinuity of compressible viscous flows grazing a non-convex corner. J. Math. Pures Appl. 100, 410–432 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. O.A. Ladyzhenskaya, V.A. Solonnikov, On some problems of vector analysis and generalized formulations of boundary value problems for the Navier-Stokes equations. Zapiski Nauchn. Sem. LOMI. 59, 81–116 (1976). English Transl.: J. Sov. Math. 10(2), 257–285 (1978)

    Google Scholar 

  34. O.A. Ladyzhenskaya, V.A. Solonnikov, On the solvability of boundary value problems for the Navier-Stokes equations in regions with noncompact boundaries. Vestnik Leningrad. Univ. 13(3), 39–47 (1977). English Transl.: Vestnik Leningrad. Univ. Math. 10, 271–280 (1982)

    Google Scholar 

  35. P.L. Lions, Compacité des solutions des équations de Navier-Stokes compressibles isentropiques. C.R. Acad. Sci. Paris 317, 115–120 (1993)

    Google Scholar 

  36. P.L. Lions, Mathematical Topics in Fluid Mechanics. Compressible Models, vol. II (Clarendon Press, Oxford, 1998)

    Google Scholar 

  37. C. Morrey, Multiple Integrals in the Calculus of Variations. Grundlehren der Mathematischen Wissenschaften, vol. 130 (Springer, Berlin/Heidelberg/New York, 1966)

    Google Scholar 

  38. F. Murat, Compacité par compensation. Annali della Scuola Normale Superiore di Pisa 5, 489–507 (1978)

    MathSciNet  MATH  Google Scholar 

  39. J. Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. Fr. 90, 487–497 (1962)

    Article  MATH  Google Scholar 

  40. S.A. Nazarov, A. Novotný, K. Pileckas, On steady compressible Navier-Stokes equations in plane domains with corners. Math. Ann. 304(1), 121–150 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Nečas, Les Methodes Directes en théorie des Équations Elliptiques (Masson and CIE, Éditeurs, Paris, 1967)

    MATH  Google Scholar 

  42. S. Novo, A. Novotný, On the existence of weak solutions to steady compressible Navier-Stokes equations when the density is not square integrable. J. Math. Kyoto Univ. 42(3), 531–550 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. S. Novo, A. Novotný, On th existence of weak solutions to the steady compressible Navier-Stokes equations in domains with conical outlets. J. Math. Fluid Mech. 8(2), 187–210 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. S. Novo, A. Novotný, M. Pokorný, Steady compressible Navier-Stokes equations in domains with non-compact boundaries. Math. Methods Appl. Sci. 28(12), 1445–1479 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Novotný, Some remarks about the compactness of steady compressible Navier-Stokes equations via the decomposition method. Comment. Math. Univ. Carolinae. 37(2), 305–342 (1996)

    MathSciNet  MATH  Google Scholar 

  46. A. Novotný, M. Padula, L p approach to steady flows of viscous compressible fluids in exterior domains. Arch. Ration. Mech. Anal. 126, 243–297 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  47. A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow (Oxford University Press, Oxiford, 2004)

    MATH  Google Scholar 

  48. M. Padula, Existence and uniqueness for viscous steady compressible motions. Arch. Ration. Mech. Anal. 77, 89–102 (1987)

    MathSciNet  MATH  Google Scholar 

  49. T. Piasecki, Steady compressible Navier-Stokes flow in a square. J. Math. Anal. Appl. 357, 447–467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. T. Piasecki, On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain. J. Differ. Equ. 248, 2171–2198 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. T. Piasecki, M. Pokorný, Strong solutions to the Navier-Stokes-Fourier system with slip-inflow boundary conditions. Z. Angew. Math. Mech. 94, 1035–1057 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  52. K. Pileckas, On unique solvability of boundary value problems for the Stokes system of equations in domains with noncompact boundaries. Trudy Mat. Inst. Steklov 147, 115–123 (1980). English Transl.: Proc. Mat. Inst. Steklov 147(2), 117–126

    Google Scholar 

  53. K. Pileckas, Recent advances in the theory of Stokes and Navier-Stokes equations in domains with noncompact boundaries, in Nonlinear Partial Differential Equations and Their Applications: Collège de France Seminar. Pitman Research Notes in Mathematics vol. 304 (Longman, Harlow, 1996), pp. 30–85

    Google Scholar 

  54. K. Pileckas, V.A. Solonnikov, Certain spaces of solenoidal vectors and solvability of the boundary value problem for the Navier-Stokes system of equations in domains with nonocompact boundaries. Zapiski Nauchn. Sem. LOMI 73, 136–151. English Transl.: J. Sov. Math. 34(6), 2101–2111 (1977)

    Google Scholar 

  55. P.I. Plotnikov, W. Weigant, Steady 3D viscous compressible flows with adiabatic exponent γ ∈ (1, ∞). J. Math. Pures Appl. 104, 58–82 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  56. P.I. Plotnikov, J. Sokolowski, Concentrations of solutions to time-discretizied compressible Navier-Stokes equations. Commun. Math. Phys. 258, 567–608 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. P.I. Plotnikov, J. Sokolowski, On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations. J. Math. Fluid Mech. 7, 529–573 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  58. P.I. Plotnikov, E.V. Ruban, J. Sokolowski, Inhomogeneous boundary value problems for compressible Navier-Stokes equations: well-posedness and sensitivity analysis. SIAM J. Math. Anal. 40, 1152–1200 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  59. P.I. Plotnikov, E.V. Ruban, J. Sokolowski, Inhomogeneous boundary value problems for compressible Navier-Stokes an transport equations. J. Math. Pures Appl. 92, 113–162 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  60. D. Serre, Variation de grande amplitude pour la densité d’um fluid visqueux compressible. Phys. D 48, 113–128 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  61. V.A. Solonnikov, On the solvability of boundary and initial boundary value problems for the Navier-Stokes system in domains with noncompact boundaries. Pac. J. Math. 93(2), 443–458 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  62. L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Anal and Mechanical, ed. by L. Knopps. Research Notes in Mathematics, vol. 39 (Pitman, Boston, 1975), pp. 136–211

    Google Scholar 

  63. A. Valli, On the existence of stationary solutions to compressible Navier-Stokes equations. Ann. Inst. H. Poincaré. 4, 99–113 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  64. A. Valli, W.M. Zajaczkowski, Navier-Stokes equations for compressible fluids: global existence and qualitabive properties of the solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986)

    Article  MATH  Google Scholar 

  65. A. Visintin, Strong convergence results related to strong convexity. Commun. Partial Differ. Equ. 9(5), 439–466 (1984)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, China

    Song Jiang

  2. Department of Mathematics, Southeast University, Nanjing, China

    Chunhui Zhou

Authors
  1. Song Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunhui Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Song Jiang .

Editor information

Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this entry

Cite this entry

Jiang, S., Zhou, C. (2017). Existence of Stationary Weak Solutions for the Isentropic and Isothermal Flows. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_63-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10151-4_63-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10151-4

  • Online ISBN: 978-3-319-10151-4

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation