On Some Boundary Value Problems for Flows with Shear Dependent Viscosity

  • H. Beirão da Veiga
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)


This notes concern the Navier-Stokes equations with gradient dependent viscosity and slip (or non-slip) type boundary conditions. Regularity up to the boundary still presents many open problems. In the sequel we present some regularity results for weak solutions to the Ladyzhenskaya model in the half space ℝ + n . See Theorems 3.1 and 3.2. Complete proofs of these results are done, and will appear in the forthcoming paper [6].


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  1. [1]
    H. AMANN, Stability of the rest state of a viscous incompressible fluid, Arch. Rat. Mech. Anal., 126 (1944), 231–242.CrossRefMathSciNetGoogle Scholar
  2. [2]
    A. AVANTAGGIATI, Spazi di Sobolev con peso ed alcune applicazioni, J.Fluid Mech., 30 (1967), 197–207.CrossRefGoogle Scholar
  3. [3]
    G.J. BEAVERS, D.D. JOSEPH, Boundary conditions of a naturally permeable wall, J.Fluid Mech., 30 (1967), 197–207.CrossRefGoogle Scholar
  4. [4]
    H. BEIRÃO DA VEIGA, Regularity of solutions to a nonhomogeneous boundary value problem for general Stokes systems in+n, to appear in Math. Annalen.Google Scholar
  5. [5]
    H. BEIRÃO DA VEIGA, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip type boundary conditions, to appear in Advances in Diff. Equations.Google Scholar
  6. [6]
    H. BEIRÃO DA VEIGA, On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip or non-slip boundary conditions equations, to appear in Comm. Pure Appl. Math.Google Scholar
  7. [7]
    H. BEIRÃO DA VEIGA, in preparation.Google Scholar
  8. [8]
    C. CONCA, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl., 64 (1985), 31–75.zbMATHMathSciNetGoogle Scholar
  9. [9]
    J. Frehse, J. Málek, and M. Steinhauer, An existence result for fluids with shear dependent viscosity-steady flows, Nonlinear Analysis. TMA., 30 (1997), 3041–3049.zbMATHCrossRefGoogle Scholar
  10. [10]
    H. Fujita, Remarks on the Stokes flow under slip and leak boundary conditions of friction type, in “Topics in Mathematical Fluid Mechanics”, Quaderni di Matematica, Vol.10, Napoli 2002, 73–94.Google Scholar
  11. [11]
    G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Vol.1: Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38, Second corrected printing, Springer-Verlag, 1998.Google Scholar
  12. [12]
    G.P. Galdi, W. Layton, Approximation of the larger eddies in fluid motion: A model for space filtered flow, Math. Models and Meth. in Appl. Sciences, 3 (2000), 343–350.MathSciNetGoogle Scholar
  13. [13]
    B. Hanouzet, Espaces de Sobolev avec poids. Application au problème de Dirichlet dans un demi espace, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227–272.MathSciNetGoogle Scholar
  14. [14]
    T.J.R. Hughes, L. Mazzei, and A.A. Oberai, The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence, Physics of Fluids, 13 (2001), 505–512.CrossRefGoogle Scholar
  15. [15]
    V. John, Slip with friction and penetration with resistence boundary conditions for the Navier-Stokes equations-numerical tests and aspects of the implementations, J. Comp. Appl. Math., to appear.Google Scholar
  16. [16]
    V. John, Large eddy simulation of turbulent incompressible flows. Analytical and numerical results for a class of LES models, Habilitationsschrift, Otto-von-Guericke-Universitat Magdeburg, April 2002.zbMATHGoogle Scholar
  17. [17]
    O.A. Ladysenskaya, On nonlinear problems of continuum mechanics, Proc. Int. Congr. Math. (Moscow, 1966). Nauka, Moscow, 1968, p.p.560–573; English transl. in Amer.Math. Soc. Transl. (2), 70 (1968).Google Scholar
  18. [18]
    O.A. Ladyzenskaya, Sur de nouvelles équations dans la dynamique des fluides visqueux et leurs résolution globale, Troudi Math. Inst. Steklov, CII (1967), 85–104.Google Scholar
  19. [19]
    O.A. Ladysenskaya, Sur des modifications des equations de Navier-Stokes pour des grand gradients de vitesses, Séminaire Inst. Steklov, 7 (1968), 126–154.Google Scholar
  20. [20]
    O.A. Ladysenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New-York, 2o édition, 1969.Google Scholar
  21. [21]
    O.A. Ladyzenskaya, G.A. Seregin, On regularity of solutions to two-dimensional equations of the dynamics of fluids with nonlinear viscosity, Zap. Nauch. Sem. Pt. Odel. Mat. Inst., 259 (1999), 145–166.Google Scholar
  22. [22]
    A. Liakos, Discretization of the Navier-Stokes equations with slip boundary condition, Num. Meth. for Partial Diff. Eq., 1 (2001), 1–18.MathSciNetGoogle Scholar
  23. [23]
    J.-L. Lions, Sur cértaines équations paraboliques non linéaires, Bull. Soc. Math. France, 93 (1965), 155–175.zbMATHMathSciNetGoogle Scholar
  24. [24]
    J.-L. Lions, Quelques Méthodes de Resolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.zbMATHGoogle Scholar
  25. [25]
    J. Malek, J. Nečas, and M. Ružička, On weak solutions to a class of non-Newtonian incompressilble fluids in bounded three-dimensional domains: the case p ≥ 2, Advances in Diff. Equations, 6 (2001), 257–302.zbMATHGoogle Scholar
  26. [26]
    J. Malek, K.R. Rajagopal, and M. Ružička, Existence and regulaity of solutions and stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci., 6, (1995), 789–812.CrossRefGoogle Scholar
  27. [27]
    C. Pare’s, Existence, uniqueness and regularity of solutions of the equations of a turbulence model for incompressible fluids, Appl. Analysis, 43 (1992), 245–296.MathSciNetGoogle Scholar
  28. [28]
    M. Ružičza, A note on steady flow of fluids with shear dependent viscosity, Nonlinear Analysis. TMA., 30 (1997), 3029–3039.CrossRefGoogle Scholar
  29. [29]
    H. Saito, L.E. Scriven, Study of the coating flow by the finite elemente method, J. Comput. Phys., 42 (1981), 53–76.zbMATHCrossRefGoogle Scholar
  30. [30]
    G.A. Seregin, Interior regularity for solutions to the modified Navier-Stokes equations, J.Math. Fluid Mech., 1 (1999), 235–281.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    J. Serrin, Mathematical Principles of Classical Fluid Mechanics, in Encyclopedia of Physics VIII, p.p. 125–263, Springer-Verlag, Berlin, 1959.Google Scholar
  32. [32]
    J. Silliman, L.E. Scriven, Separating flow near a static contact line: slip at a wall and shape of a free surface, J. Comput. Physics, 34 (1980), 287–313.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    C.G. Simader, H. Sohr, The Dirichlet problem for the Laplacian in Bounded and unbounded domains, Pitman Research Notes in Mathematics Series., Longman Scientific and Technical, 360, 1997.Google Scholar
  34. [34]
    J.S. Smagorinsky, General circulation experiments with the primitive equations. I. The basic experiment, Mon. Weather Rev., 91 (1963), 99–164.Google Scholar
  35. [35]
    V.A. Solonnikov and V.E. Ščadilov, On a boundary value problem for a stationary system of Navier-Stokes equations, Proc. Steklov Inst. Math., 125 (1973), 186–199.zbMATHGoogle Scholar
  36. [36]
    G. Stokes, Trans. Cambridge Phil. Soc, 8, 287 (1845), 75–129.Google Scholar
  37. [37]
    R. Verfűrth, Finite element approximation of incompressible Navier-Stokes equations with slip boundary conditions, Numer. Math., 50 (1987), 697–721.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • H. Beirão da Veiga
    • 1
  1. 1.Dept. of Applied Mathematics “U. Dini,”University of PisaPisaItaly

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