Skip to main content
SpringerLink
Log in

Weak–Strong Uniqueness for the Isentropic Compressible Navier–Stokes System

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

We prove weak–strong uniqueness results for the isentropic compressible Navier–Stokes system on the torus. In other words, we give conditions on a weak solution, such as the ones built up by Lions (Compressible Models, Oxford Science, Oxford, 1998), so that it is unique. It is of fundamental importance since uniqueness of these solutions is not known in general. We present two different methods, one using relative entropy, the other one using an improved Gronwall inequality due to the author; these two approaches yield complementary results. Known weak–strong uniqueness results are improved and classical uniqueness results for this equation follow naturally.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berthelin F., Vasseur A.: From kinetic equations to multidimensional isentropic gas dynamics before shocks. SIAM J. Math. Anal. 36(6), 1807–1835 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cho Y., Choe H.J., Kim H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. 83(9), 243–275 (2004)

    MATH  MathSciNet  Google Scholar 

  3. Choe H.J., Kim H.: Strong solutions of the Navier–Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190(2), 504–523 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Danchin R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141(3), 579–614 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Dafermos C.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70(2), 167–179 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Desjardins B.: Regularity of weak solutions of the compressible isentropic Navier–Stokes equations. Comm. Partial Differ. Equ. 22(5–6), 977–1008 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Germain P.: Multipliers, paramultipliers, and weak–strong uniqueness for the Navier–Stokes equations. J. Differ. Equ. 226(2), 373–428 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Germain, P.: Strong solutions and weak–strong uniqueness for the Navier–Stokes equation. J. Anal. Math. (accepted)

  10. Hoff D.: Spherically symmetric solutions of the Navier–Stokes equations for compressible, isothermal flow with large, discontinuous initial data. Indiana Univ. Math. J. 41(4), 1225–1302 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hoff D.: Uniqueness of weak solutions of the Navier–Stokes equations of multidimensional, compressible flow. SIAM J. Math. Anal. 37(6), 1742–1760 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hoff D.: Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Comm. Pure Appl. Math. 55(11), 1365–1407 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Itaya N.: On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluid. Kodai Math. Sem. Rep. 23, 60–120 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jiang S., Zhang P.: Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids. J. Math. Pures Appl. 82(9), 949–973 (2003)

    MATH  MathSciNet  Google Scholar 

  15. Lions, P.-L.: Mathematical topics in fluid mechanics, vol. 1. In: Incompressible Models. Oxford Lecture Series in Mathematics and its Applications, vol. 3. Oxford Science, Oxford (1996)

  16. Lions, P.-L.: Mathematical topics in fluid mechanics, vol. 2. In: Compressible Models. Oxford Lecture Series in Mathematics and its Applications, vol. 10. Oxford Science, Oxford (1998)

  17. Matsumura A., Nishida T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Jpn Acad. Ser. A Math. Sci. 55(9), 337–342 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  18. Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)

    MATH  MathSciNet  Google Scholar 

  19. Mellet A., Vasseur A.: Existence and uniqueness of global strong solutions for one-dimensional compressible Navier–Stokes equations. SIAM J. Math. Anal. 39(4), 1344–1365 (2007/08)

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  21. Prodi G.: Un teorema di unicità per le equazioni di Navier–Stokes (Italian). Ann. Mat. Pura Appl. 48(4), 173–182 (1959)

    MATH  MathSciNet  Google Scholar 

  22. Serrin, J.: The initial value problem for the Navier–Stokes equations. 1963 Nonlinear Problems Proc. Sympos., Madison, Wis. pp. 69–98. University of Wisconsin Press, Madison

  23. Weigant, V.A.: An example of the nonexistence with respect to time of the global solution of Navier–Stokes equations for a compressible viscous barotropic fluid. (Russian) Dokl. Akad. Nauk 339(2), 155–156 (1994); translation in Russian Acad. Sci. Dokl. Math. 50(3), 397–399 (1995)

Download references

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012-1185, USA

    Pierre Germain

Authors
  1. Pierre Germain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Germain.

Additional information

Communicated by D. Chae

Rights and permissions

Reprints and permissions

About this article

Cite this article

Germain, P. Weak–Strong Uniqueness for the Isentropic Compressible Navier–Stokes System. J. Math. Fluid Mech. 13, 137–146 (2011). https://doi.org/10.1007/s00021-009-0006-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-009-0006-1

Keywords

Search

Navigation