Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical Cases

  • P. I. Plotnikov
  • W. Weigant
Living reference work entry


In this chapter the compressible Navier-Stokes equations with the critical adiabatic exponents are considered. The crucial point in this situation are new estimates of the Radon measure of solutions. These estimates are applied to the boundary value problem for the compressible Navier-Stokes equations with the critical adiabatic exponents. The existence of weak solutions to 2D isothermal problem is proved. The cancelation of concentrations for 3D nonstationary initial-boundary value problem with the critical adiabatic exponent 3/2 is established.



This work was supported by Russian Science Foundation, project 15-11-20019.


  1. 1.
    R.A. Adams, Sobolev Spaces (Academic press, New-York, 1975)MATHGoogle Scholar
  2. 2.
    B. Ducomet, S. Nečasová, A. Vasseur, On spherically symmetric motions of a viscous compressible barotropic and selfgravitating gas. J. Math. Fluid Mech. 99, 1–24 (2009)MATHGoogle Scholar
  3. 3.
    L. C. Evans, Partial Differential Equations (Am. Math. Soc., Providence, RI, 1998)MATHGoogle Scholar
  4. 4.
    E. Feireisl, Dynamics of Viscous Compressible Fluids (Oxford University Press, Oxford, 2004)MATHGoogle Scholar
  5. 5.
    E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3(3), 358–392 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    S. Jiang, P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Commun. Math. Phys. 215, 559–581 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    S. Jiang, P. Zhang, Axisymmetric solutions to the 3D Navier-Stokes equations for compressible isentropic fluids. J. Math. Pures Appl. 82, 949–973 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    P.L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models (Clarendon Press, Oxford, 1998)Google Scholar
  9. 9.
    P.L. Lions, On some chalenging problems in nonlinear partial differential equations, in Mathematics: Frontiers and Perspectives, ed. by V. Arnold, M. Atyah, P. Lax, D. Mazur (AMS, Providence, 2000)Google Scholar
  10. 10.
    A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and Its Applications, vol. 27 (Oxford University Press, Oxford, 2004)Google Scholar
  11. 11.
    M. Padula, Existence of global solutions foe two-dimensional viscous compressible flows. J. Funct. Anal. 69(1), 1–20 (1986)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    M. Padula, Correction. J. Funct. Anal. 76(1), 70–76 (1988)MathSciNetGoogle Scholar
  13. 13.
    P.I. Plotnikov, J. Sokolowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization (Birkhauser, Basel, 2012)Google Scholar
  14. 14.
    P.I. Plotnikov, W. Weigant, Isothermal Navier-Stokes equations and radon transform. SIAM J. Math. Anal. 47, 626–652 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    P.I. Plotnikov, W. Weigant, Rotationally Symmetric Viscous Gas Flows. Computational mathematics and mathematical physics 57, 387–400 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Mathematical DepartmentNovosibirsk State UniversityNovosibirskRussia
  2. 2.Siberian Division of Russian Academy of SciencesLavryentyev Institute of HydrodynamicsNovosibirskRussia
  3. 3.Institute für Angewandte MathematikUniversität BonnBonnGermany

Personalised recommendations