Evolution problems of Navier–Stokes type with anisotropic diffusion

Original Paper

Abstract

In this work, we consider the evolutive problem for the incompressible Navier–Stokes equations with a general diffusion which can be fully anisotropic. The existence of weak solutions is proved for the associated initial problem supplemented with no-slip boundary conditions. We prove also the properties of extinction in a finite time, exponential time decay and power time decay. With this respect, we consider the important case of a forces fields with possible different behavior in distinct directions. Perturbations of the asymptotically stable equilibrium are established as well.

Keywords

Anisotropic diffusion Navier–Stokes Existence  Finite time extinction Exponential time decay Power time decay 

Mathematics Subject Classification

35Q35 76D05 35Q30 76D03 35D30 35B40 

References

  1. 1.
    Antontsev, S.N., Díaz, J.I., Shmarev, S.I.: Energy methods for free boundary problems. In: Progress in Nonlinear Differential Equations, vol. 48. Birkhäuser, Boston (2002)Google Scholar
  2. 2.
    Antontsev, S., Chipot, M.: Anisotropic equations: uniqueness and existence results. Differ. Integral Equ. 21(5–6), 401–419 (2008)MathSciNetMATHGoogle Scholar
  3. 3.
    Antontsev, S.N.,  de Oliveira, H.B.: Finite time localized solutions of fluid problems with anisotropic dissipation. In: International Series of Numerical Mathematics, vol. 154, pp. 23–32. Birkhäuser, Boston (2006)Google Scholar
  4. 4.
    Antontsev, S.N., de Oliveira, H.B.: Analysis of the existence for the steady Navier–Stokes equations with anisotropic diffusion. Adv. Differ. Equ. 19(5–6), 441–472 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Antontsev, S.N., Shmarev, S.I.: Localization of solutions of anisotropic parabolic equations. Nonlinear Anal. 71(12), 725–737 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Fluids with anisotropic viscosity. Special issue for R. Temam’s 60th birthday. M2AN Math. Model. Numer. Anal. 34(2), 315–335 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chemin, J.-Y., Zhang, P.: On the global wellposedness to the 3-D incompressible anisotropic Navier–Stokes equations. Commun. Math. Phys. 272(2), 529–566 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Diening, L., Ru̇žička, M., Wolf, J.: Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5(IX), 1–46(2010)Google Scholar
  9. 9.
    Fragalà, I., Gazzola, F., Kawohl, B.: Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(5), 715–734 (2004)Google Scholar
  10. 10.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. 2. Nonlinear Steady Problems. Springer, Berlin (1994)Google Scholar
  11. 11.
    Haškovec, J., Schmeiser, C.: A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems. Monatsh. Math. 158(1), 71–79 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Iftimie, D.: The resolution of the Navier–Stokes equations in anisotropic spaces. Rev. Mat. Iberoam. 15(1), 1–36 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, Science Publishers, New York (1969)MATHGoogle Scholar
  15. 15.
    Lions, J.-L.: Quelques mèthodes de résolution des problèmes aux limites non liniaires. Dunod, Paris (1969)MATHGoogle Scholar
  16. 16.
    Málek, J., Nečas, J., Rokyta, M., Ru̇žička, M.: Weak and measure-valued solutions to evolutionary PDEs. In: Applied Mathematics and Mathematical Computation, vol. 13. Chapman & Hall, London (1996)Google Scholar
  17. 17.
    Penel, P., Pokorný, M.: Improvement of some anisotropic regularity criteria for the Navier–Stokes equations. Discrete Contin. Dyn. Syst. Ser. S 6(5), 1401–1407 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Porzio, M.M.: \(\rm L^{\infty }\)-regularity for degenerate and singular anisotropic parabolic equations. Boll. Un. Mat. Ital. A 11(7), 697–707 (1997)MathSciNetMATHGoogle Scholar
  19. 19.
    Rákosník, J.: Some remarks to anisotropic Sobolev spaces. I. Beiträge Anal. No. 15(1979), 55–68 (1979)Google Scholar
  20. 20.
    Rákosník, J.: Some remarks to anisotropic Sobolev spaces. II. Beiträge Anal. No. 15(1980), 127–140 (1981)Google Scholar
  21. 21.
    Starovoitov, V.N., Tersenov, A.S.: Singular and degenerate anisotropic parabolic equations with a nonlinear source. Nonlinear Anal. 72(6), 3009–3027 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1979)MATHGoogle Scholar
  23. 23.
    Troisi, M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ric. Mat. 18, 3–24 (1969)MathSciNetMATHGoogle Scholar
  24. 24.
    Vétois, J.: Existence and regularity for critical anisotropic equations with critical directions. Adv. Differ. Equ. 16(1–2), 61–83 (2011)MathSciNetMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zhikov, V.: New approach to the solvability of generalized Navier–Stokes equations. Funct. Anal. Appl. 43(3), 190–207 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.CMAFCIO, Universidade de LisboaLisbonPortugal
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.FCT, Universidade do AlgarveFaroPortugal

Personalised recommendations