Existence of weak solutions for the generalized Navier–Stokes equations with damping

Article

Abstract

In this work we consider the generalized Navier–Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier–Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any \({q > \frac{2N}{N+2}}\) and any σ > 1, where q is the exponent of the diffusion term and σ is the exponent which characterizes the damping term.

Mathematics Subject Classification (2010)

35D05 35K55 35Q30 76D03 76D05 

Keywords

Generalized Navier–Stokes Damping Existence of weak solutions Decomposition of the pressure Lipschitz truncation 

References

  1. 1.
    Antontsev S.N., Díaz J.I., de Oliveira H.B.: On the confinement of a viscous fluid by means of a feedback external field. C. R. Méc. Acad. Sci. Paris 330, 797–802 (2002)MATHCrossRefGoogle Scholar
  2. 2.
    Antontsev S.N., Díaz J.I., de Oliveira H.B.: Stopping a viscous fluid by a feedback dissipative field. I. The stationary Stokes problem. J. Math. Fluid Mech. 6(4), 439–461 (2004)MathSciNetMATHGoogle Scholar
  3. 3.
    Antontsev S.N., Díaz J.I., de Oliveira H.B.: Stopping a viscous fluid by a feedback dissipative field. II. The stationary Navier–Stokes problem. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl 15(3–4), 257–270 (2004)MathSciNetMATHGoogle Scholar
  4. 4.
    Antontsev S.N., de Oliveira H.B.: The Navier–Stokes problem modified by an absorption term. Appl. Anal. 89(12), 1805–1825 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Antontsev S.N., de Oliveira H.B.: The Oberbeck-Boussinesq problem modified by thermo-absorption term. J. Math. Anal. Appl. 379(2), 802–817 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Barret J.W., Liu W.B.: Finite element approximation of the parabolic p-laplacian. SIAM J. Numer. Anal. 31(2), 413–428 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bae H.-O.: Existence, regularity, and decay rate of solutions of non-Newtonian flow. J. Math. Anal. Appl. 231(2), 467–491 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Benilan Ph., Brezis H., Crandall M.: A semilinear equation in L 1(R N). Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(4), 523–555 (1975)MathSciNetMATHGoogle Scholar
  9. 9.
    Bernis F.: Elliptic and parabolic semilinear problems without conditions at infinity. Arch. Ration. Mech. Anal. 106, 217–241 (1989)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Cai X., Jiu Q.: Weak and strong solutions for the incompressible Navier–Stokes equations with damping. J. Math. Anal. Appl. 343(2), 799–809 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Carles R., Gallo C.: Finite time extinction by nonlinear damping for the Schrodinger equation. Commun. Partial Differ. Equ. 36(6), 961–975 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Diening L., Ru̇žička M., Wolf J.: Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Scuola Norm. Sup. Pisa Cl. Csi. 5(IX), 1–46 (2010)Google Scholar
  13. 13.
    Frehse J., Málek J., Steinhauer M.: An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear Anal. 30(5), 3041–3049 (1997)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Frehse J., Málek J., Steinhauer M.: On analysis of steady flows of fluids with shear dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34(5), 1064–1083 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations. Linearized steady problems, vol. I. Springer Tracts in Natural Philosophy, vol. 38. Springer, New York (1994)Google Scholar
  16. 16.
    Ladyzhenskaya O.A.: New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problem for them. Proc. Steklov Inst. Math. 102, 95–118 (1967)Google Scholar
  17. 17.
    Lions J.L.: Quelques mèthodes de résolution des problèmes aux limites non liniaires. Dunod, Paris (1969)Google Scholar
  18. 18.
    de Oliveira, H.B.: On the influence of an absorption term in incompressible fluid flows. In: Advances in Mathematical Fluid Mechanics, pp. 409–424. Springer, Berlin (2010)Google Scholar
  19. 19.
    Pan R., Zhao K.: The 3D compressible Euler equations with damping in a bounded domain. J. Differ. Equ. 246(2), 581–596 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Simader, C.G., Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in Lq-spaces for bounded and exterior domains. In: Mathematical Problems Relating to the Navier–Stokes Equation, pp. 1–35. Ser. Adv. Math. Appl. Sci., vol. 11. World Scientific, River Edge (1992)Google Scholar
  21. 21.
    Simon J.: Compact sets in the space L p(0, T;B). Ann. Mat. Pura Appl. 146(4), 65–96 (1987)MathSciNetMATHGoogle Scholar
  22. 22.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)Google Scholar
  23. 23.
    Truesdell C.: Precise theory of the absorption and dispersion of forced plane infinitesimal waves according to the Navier–Stokes equations. J. Ration. Mech. Anal. 2, 643–741 (1953)MathSciNetMATHGoogle Scholar
  24. 24.
    Wolf J.: Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9(1), 104–138 (2007)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Zhikov V.: New approach to the solvability of generalized Navier–Stokes equations. Funct. Anal. Appl. 43(3), 190–207 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhou Y.: Global existence and nonexistence for a nonlinear wave equation with damping and source terms. Math. Nachr. 278(11), 1341–1358 (2005)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.FCTUniversidade do AlgarveFaroPortugal
  2. 2.CMAFUniversidade de LisboaLisbonPortugal

Personalised recommendations