Turbulence models, p-fluid flows, and W2,L regularity of solutions

Communications on Pure and Applied Analysis

Volumn 8, Issue 2, 2009, Pages 769-783

  Da Veiga, Hugo Beirão ^a  

^a UNIVERSITY OF PISA   (Italy)

Author keywords

Boundary value problems; Navier Stokes equations; Regularity; Shear dependent viscosity

Indexed keywords

EID: 63349103115     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2009.8.769     Document Type: Article

References (27)

1. H. Beirão da Veiga, On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip and non-slip boundary conditions, Comm. Pure Appl. Math., 58 (2005), 552-577.
2. H. Beirão da Veiga, Navier-Stokes equations with shear thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 9 (2007), 506-516.
3. H. Beirão da Veiga, On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc, in press.
4. H. Beirão da Veiga, On the global regularity of shear thinning flows in smooth domains, J. Math. Anal. Appl. (2008), doi: 10.1016/j.jmaa.2008. 09.009.
5. H. Beirão da Veiga, P. Kaplický and M. Růž ička, Boundary regularity of shear thickening flows, to appear.
6. 2, q-regularity of incompressible fluids with shear-dependent viscosities: The shear-thinning case, J. Math. Fluid Mech., DOI 10.1007/s00021-008-0254-5 (2008), in press.
7. F. Crispo, On the regularity of shear-thickening viscous fluids, Chin. Ann. of Math., in press.
8. G. P. Galdi, Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics, in "Hemodynamical Flows: Modeling, Analysis and Simulation" (eds. G. P. Galdi, A. M. Robertson, R. Rannacher and S. Turek), Oberwolfach Seminars, Birkhauser-Verlag, 2007.
9. 1, α-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 259 (1999), no.
10. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 30, 89-121, 297.
11. O. A. Ladyzhenskaya, On nonlinear problems of continuum mechanics, "Proc. Int. Congr. Math., Moscow, 1966", 560-573. Nauka, Moscow, 1968.
12. English transl. in Amer. Math. Soc. Transl., (2) 70, 1968.
13. O. A. Ladyhzenskaya, Sur de nouvelles équations dans la dynamique des fluides visqueux et leurs résolution globale, Troudi Math. Inst. Steklov, CII (1967), 85-104.
14. O. A. Ladyzhenskaya, Sur des modifications des équations de Navier-Stokes pour des grand gradients de vitesses, Séminaire Inst. Steklov, 7 (1968), 126-154.
15. O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow", Second edition. Gordon and Breach, New-York, 1969.
16. J. L. Lions, Sur cértaines équations paraboliques non linéaires, Bull. Soc. Math. France, 93 (1965), 155-175.
17. J. L. Lions, "Quelques Méhodes de Résolution des Problèmes aux Limites Non Linéaires", Dunod, Paris, 1969.
18. J. Málek, J. Nečas and M. Ružicka, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2, Advances in Diff. Equations, 6 (2001), 257-302.
19. J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in "Evolutionary Equations. Vol. II", Handb. Differ. Equ. series, Elsevier/North-Holland, Amsterdam, (2005), 371-459.
20. J. Málek, 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.
21. J. Nečas, "Équations aux Dérivées Partielles", Presses de l'Université de Montréal, Montréal, 1965.
22. C. Parés, Existence, uniqueness and regularity of solutions of the equations of a turbulence model for incompressible fluids, Appl. Analysis, 43 (1992), 245-296.
23. K. R. Rajagopal, Mechanics of Non-Newtonian Fluids, Recent Developments in Theoretical Fluid Mechanics (G. P. Galdi and J. Nečas, eds.), Research Notes in Mathematics Series, vol. 291, Longman, 1993, pp. 129-162.
24. J. Serrin, Mathematical principles of classical fluid mechanics, Encyclopedia of Physics VIII, 125-263. Springer-Verlag, Berlin, 1959.
25. J. S. Smagorinsky, General circulation experiments with the primitive equations. I. The basic experiment, Mon. Weather Rev., 91 (1963), 99-164.
26. G. Stokes, Trans. Cambridge Phil. Soc, 8, 287 (1845), 75-129.
27. M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche di Matematica, 18 (1969), 3-24.