On the regularity of flows with ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions

Communications on Pure and Applied Mathematics

Volumn 58, Issue 4, 2005, Pages 552-577

  Beirao Da Veiga H ^a  

^a UNIVERSITY OF PISA   (Italy)

Author keywords

[No Author keywords available]

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EID: 15544361893     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.20036     Document Type: Article

References (55)

1. Amann, H. Stability of the rest state of a viscous incompressible fluid. Arch. Rational Mech. Anal. 126 (1994), no. 3, 231-242.
2. Avantaggiati, A. Spazi di Sobolev con peso ed alcune applicazioni. Boll. Un. Mat. Ital. (5) 13A (1976), no. 1, 1-52.
3. Beavers, G. S.; Joseph, D. D. Boundary conditions at a naturally permeable wall. I. J. Fluid Mech. 30 (1967), 197-207.
4. Beirão da Veiga, H. Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip type boundary conditions. Adv. Differential Equations 9 (2004), no.9-10, 1079-1114.
5. n. Math. Ann., in press.
6. Beirão da Veiga, H. On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip and non-slip boundary conditions. Part II: The evolution problem. In preparation.
7. Bellout, H.; Bloom, F.; Nečas, J. Young measure-valued solutions for non-Newtonian incompressible fluids. Comm. Partial Differential Equations 19 (1994), no. 11-12, 1763-1803.
8. Conca, C. On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl. (9) 64 (1985), no. 1, 31-75.
9. Consiglieri, L. Weak solutions for a class of non-Newtonian fluids with energy transfer. J. Math. Fluid Mech. 2 (2000), no. 3, 267-293.
10. Diening, L.; Prohl, A.; Růžička, M. On time-discretizations for generalized Newtonian fluids. Nonlinear problems in mathematical physics and related topics, II, 89-118. International Mathematical Series (New York), 2. Kluwer/Plenum, New York, 2002.
11. Frehse, J.; Málek, J. Problems due to the no-slip boundary in incompressible fluid dynamics. Geometric analysis and nonlinear partial differential equations, 559-571. Springer, Berlin, 2003.
12. Frehse, J.; Málek, J.; Steinhauer, M. An existence result for fluids with shear dependent viscosity - steady flows. Nonlinear Anal. 30 (1997), 3041-3049.
13. Fuchs, M.; Seregin, G. Some remarks on non-Newtonian fluids including nonconvex perturbations of the Bingham and Powell-Eyring model for viscoplastic fluids. Math. Models Methods Appl. Sci. 7 (1997), no. 3, 405-433.
14. Fujita, H. Remarks on the Stokes flow under slip and leak boundary conditions of friction type. Topics in mathematical fluid mechanics (Napoli, 2000), 73-94. Quaderni di Matematica, 10. Napoli, 2002.
15. Galdi, G. P. An introduction to the mathematical theory of the Navier-Stokes equations. I. Linearized steady problems. 2nd ed. Springer Tracts in Natural Philosophy, 38. Springer, Berlin-New York, 1998.
16. Galdi, G. P.; Layton, W. J. Approximation of the larger eddies in fluid motions. II. A model for space-filtered flow. Math. Models Methods Appl. Sci. 10 (2000), no. 3, 343-350.
17. Guo, B.; Zhu, P. Partial regularity of suitable weak solutions to the system of the incompressible non-Newtonian fluids. J. Differential Equations 178 (2002), no. 2, 281-297.
18. Hanouzet, B. Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46 (1971), 227-272.
19. Hughes, T. J. R.; Mazzei, L.; Oberai, A. A.; Wray, A. The multiscale formulation of large eddy simulation: decay of homogeneous Isotropic turbulence. Phys. Fluids 13 (2001), no. 2, 505-512.
20. John, V. Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations - numerical tests and aspects of the implementation. J. Comput. Appl. Math. 147 (2002), no. 2, 287-300.
21. John, V. Large eddy simulation of turbulent incompressible flows. Analytical and numerical results for a class of LES models. Lecture Notes in Computational Science and Engineering, 34. Springer, Berlin, 2004.
22. Ladyhzenskaya, O. A. Sur de nouvelles équations dans la dynamique des fluides visqueux et leurs résolution globale. Trudy Mat. Inst. Steklov. 102 (1967), 85-104.
23. Ladyzhenskaya, O. A. On nonlinear problems of continuum mechanics. Proc. Int. Congr. Math. (Moscow, 1966), 560-573. Nauka, Moscow, 1968; translation in Amer. Math. Soc. Transl. (2) 70, 1968.
24. Ladyzhenskaya, O. A. On nonlinear problems of continuum mechanics. Proc. Int. Congr. Math. (Moscow, 1966), 560-573. Nauka, Moscow, 1968; translation in Amer. Math. Soc. Transl. (2) 70, 1968.
25. Ladyzhenskaya, O. A. Sur des modifications des équations de Navier-Stokes pour des grand gradients de vitesses. Sem. Math. V A. Steklov 7 (1968), 126-154.
26. Ladyzhenskaya, O. A. The mathematical theory of viscous incompressible flow. 2nd ed. Mathematics and Its Applications, 2. Gordon and Breach, New York-London-Paris, 1969.
27. Ladyzhenskaya, O. A.; Seregin, G. A. On the smoothness of systems describing flows of generalized Newtonian fluids, and on the estimation of the dimensions of their attractors. Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 1, 59-122; translation in Izv. Math. 62 (1998), no. 1, 55-113.
28. Ladyzhenskaya, O. A.; Seregin, G. A. On the smoothness of systems describing flows of generalized Newtonian fluids, and on the estimation of the dimensions of their attractors. Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 1, 59-122; translation in Izv. Math. 62 (1998), no. 1, 55-113.
29. Ladyzhenskaya, O. A.; Seregin, G. A. On the regularity of solutions of two-dimensional equations of the dynamics of fluids with nonlinear viscosity. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 259 (1999), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Tear. Funkts. 30, 145-166, 298.
30. Ladyzhenskaya, O. A.; Seregin, G. A. On the regularity of solutions of two-dimensional equations of the dynamics of fluids with nonlinear viscosity. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 259 (1999), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Tear. Funkts. 30, 145-166, 298.
31. Liakos, A. Discretization of the Navier-Stokes equations with slip boundary condition. Numer. Methods Partial Differential Equations 17 (2001), no. 1, 26-42.
32. Lions, J.-L. Problèmes aux limites dans les èquations aux dérivées partielles. 2nd éd. Séminaire de Mathématiques Supérieures, 1 (Été, 1962). University of Montreal, Montreal, 1965.
33. Lions, J.-L. Sur certaines équations paraboliques non linéaires. Bull. Soc. Math. France 93 (1965), 155-175.
34. Lions, J.-L. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris, 1969.
35. Málek, J.; Nečas, J.; Rajagopal, K. R. Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165 (2002), no. 3, 243-269.
36. Málek, J.; Nečas, J.; Rokyta, M.; Růžič ka, M. Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computation, 13. Chapman and Hall, London, 1996.
37. Málek, J.; Nečas, J.; Růžička, M. On the non-Newtonian incompressible fluids. Math. Models Methods Appl. Sci. 3 (1993), no. 1, 35-63.
38. Málek, J.; Nečas, J.; Růžička, M. On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2. Adv. Differential Equations 6 (2001), no. 3, 257-302.
39. Málek, J.; Rajagopal, K. R.; Růzicka, M. Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods Appl. Sci. 5 (1995), no. 6, 789-812.
40. Nečas, J. Équations aux dérivées partielles. University of Montreal, Montreal, 1965.
41. Nirenberg, L. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162.
42. Parés, C. Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids. Appl. Anal. 43 (1992), no. 3-4, 245-296.
43. Pokorný, M. Cauchy problem for the non-Newtonian viscous incompressible fluid. Appl. Math. 41 (1996), no. 3, 169-201.
44. Růžička, M. A note on steady flow of fluids with shear dependent viscosity. Nonlinear Anal. 30 (1997), no. 5, 3029-3039.
45. Růžička, M. Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, 1748. Springer, Berlin, 2000.
46. Saito, H; Scriven, L. E. Study of the coating flow by the finite element method. J. Comput. Phys. 42 (1981), 53-76.
47. Seregin, G. A. Interior regularity for solutions to the modified Navier-Stokes equations. J. Math. Fluid Mech. 1 (1999), no. 3, 235-281.
48. Serrin, J. Mathematical principles of classical fluid mechanics. Handbuch der Physik (herausgegeben von S. Flügge), 8/1. Strömungsmechanik I (Mitherausgeber C. Truesdell), 125-263. Springer, Berlin-Göttingen- Heidelberg, 1959.
49. Silliman, W. J.; Scriven, L. E. Separating flow near a static contact line: slip at a wall and shape of a free surface. J. Comput. Phys. 34 (1980), no. 3, 287-313.
50. Simader, C. G.; Sohr, H. The Dirichlet problem for the Laplacian in bounded and unbounded domains. A new approach to weak, strong and (2+k)-solutions in Sobolev-type spaces. Pitman Research Notes in Mathematics Series, 360. Longman, Harlow, 1996.
51. Smagorinsky, J. S. General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev. 91 (1963), no. 3, 99-164.
52. Solonnikov, V. A.; Ščadilov, V. E. A certain boundary value problem for the stationary system of Navier-Stokes equations. Boundary Value Problems of Mathematical Physics, 8. Trudy Mat. Inst. Steklov. 125 (1973), 196-210; translation in Proc. Steklov Inst. Math. 125 (1973), 186-199.
53. Solonnikov, V. A.; Ščadilov, V. E. A certain boundary value problem for the stationary system of Navier-Stokes equations. Boundary Value Problems of Mathematical Physics, 8. Trudy Mat. Inst. Steklov. 125 (1973), 196-210; translation in Proc. Steklov Inst. Math. 125 (1973), 186-199.
54. Stokes, G. Trans. Cambridge Phil. Soc., 8, 287, (1845), 75-129.
55. Verfürth, R. Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition. Numer. Math. 50 (1987), no. 6, 697-721.