Boundary regularity of flows under perfect slip boundary conditions

 Access to full text

 Full (PDF)

1. [1] Amrouche C., Girault V., Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 1994, 44(119)(1), 109–140 Zbl0823.35140
2. [2] Beirão da Veiga H., On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math., 2005, 58(4), 552–577 http://dx.doi.org/10.1002/cpa.20036[Crossref] Zbl1075.35045
3. [3] Beirão da Veiga H., Navier-Stokes equations with shear-thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 2009, 11(2), 233–257 http://dx.doi.org/10.1007/s00021-008-0257-2[Crossref][WoS] Zbl1213.76047
4. [4] Beirão da Veiga H., Navier-Stokes equations with shear-thinning viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 2009, 11(2), 258–273 http://dx.doi.org/10.1007/s00021-008-0258-1[Crossref][WoS] Zbl1190.35174
5. [5] Beirão da Veiga H., On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc. (JEMS), 2009, 11(1), 127–167 http://dx.doi.org/10.4171/JEMS/144[Crossref] 
6. [6] Beirão da Veiga H., Kaplický P., Růžička M., Boundary regularity of shear thickening flows, J. Math. Fluid Mech., 2011, 13(3), 387–404 http://dx.doi.org/10.1007/s00021-010-0025-y[Crossref] Zbl1270.35360
7. [7] Desvillettes L., Villani C., On a variant of Korn’s inequality arising in statistical mechanics, ESIAM Control Optim. Calc. Var., 2002, 8, 603–619 http://dx.doi.org/10.1051/cocv:2002036[Crossref] Zbl1092.82032
8. [8] Diening L., Ettwein F., Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., 2008, 20(3), 523–556 http://dx.doi.org/10.1515/FORUM.2008.027[WoS][Crossref] Zbl1188.35069
9. [9] Diening L., Kaplický P., L q theory for a generalized Stokes system, Manuscripta Math. (in press), DOI: 10.1007/s00229-012-0574-x [Crossref][WoS] Zbl1263.35175
10. [10] Diening L., Růžička M., Interpolation operators in Orlicz-Sobolev spaces, Numer. Math., 2007, 107(1), 107–129 http://dx.doi.org/10.1007/s00211-007-0079-9[Crossref][WoS] Zbl1131.46023
11. [11] Diening L., Růžička M., Schumacher K., A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math., 2010, 35(1), 87–114 http://dx.doi.org/10.5186/aasfm.2010.3506[Crossref] 
12. [12] Ebmeyer C., Regularity in Sobolev spaces of steady flows of fluids with shear-dependent viscosity, Math. Methods Appl. Sci., 2006, 29(14), 1687–1707 http://dx.doi.org/10.1002/mma.748[Crossref] Zbl1124.35053
13. [13] Frehse J., Málek J., Steinhauer M., An existence result for fluids with shear dependent viscosity - steady flows, Nonlinear Anal., 1997, 30(5), 3041–3049 http://dx.doi.org/10.1016/S0362-546X(97)00392-1[Crossref] Zbl0902.35089
14. [14] Frehse J., Málek J., Steinhauer M., On existence result for fluids with shear dependent viscosity - unsteady flows, In: Partial Differential Equations, Praha, August 10–16, 1998, Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, 2000, 121–129 Zbl0935.35026
15. [15] Galdi G.P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I, Springer Tracts Nat. Philos., 38, Springer, New York, 1994 Zbl0949.35004
16. [16] Hlaváček I., Nečas J., On inequalities of Korn’s type, Arch. Rational Mech. Anal., 1970, 36(4), 305–311 http://dx.doi.org/10.1007/BF00249518[Crossref] Zbl0193.39001
17. [17] Hlaváček I., Nečas J., On inequalities of Korn’s type II, Arch. Rational Mech. Anal., 1970, 36(4), 312–334 http://dx.doi.org/10.1007/BF00249519[Crossref] Zbl0193.39002
18. [18] Kaplický P., Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions, Z. Anal. Anwendungen, 2005, 24(3), 467–486 http://dx.doi.org/10.4171/ZAA/1251[Crossref] Zbl1094.35100
19. [19] Kaplický P., Regularity of flow of anisotropic fluid, J. Math. Fluid Mech., 2008, 10(1), 71–88 http://dx.doi.org/10.1007/s00021-006-0217-7[Crossref] Zbl1162.76304
20. [20] Kaplický P., Málek J., Stará J., On global existence of smooth two-dimensional steady flows for a class of non-Newtonian fluids under various boundary conditions, In: Applied Nonlinear Analysis, New York, Kluwer/Plenum, 1999, 213–229 Zbl0953.35120
21. [21] Kaplický P., Málek J., Stará J., C 1,α-solutions to a class of nonlinear fluids in two dimensions - stationary Dirichlet problem, J. Math. Sci. (New York), 2002, 109(5), 1867–1893 http://dx.doi.org/10.1023/A:1014440207817[Crossref] 
22. [22] Kaplický P., Málek J., Stará J., Global-in-time Hölder continuity of the velocity gradients for fluids with shear-dependent viscosities, NoDEA Nonlinear Differential Equations Appl., 2002, 9(2), 175–195 http://dx.doi.org/10.1007/s00030-002-8123-z[Crossref] Zbl0991.35066
23. [23] Krasnosel’skiĭ M.A., Rutickiĭ Ja.B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961 
24. [24] Ladyzhenskaya O.A., New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Proc. Steklov Inst. Math., 1967, 102, 95–118 
25. [25] Málek J., Nečas J., Rokyta M., Růžička M., Weak and Measure-Valued Solutions to Evolutionary PDEs, Appl. Math. Math. Comput., 13, Chapman & Hall, London, 1996 Zbl0851.35002
26. [26] 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, 2001, 6(3), 257–302 Zbl1021.35085
27. [27] Málek J., Pražák D., Steinhauer M., On the existence and regularity of solutions for degenerate power-law fluids, Differential Integral Equations, 2006, 19(4), 449–462 Zbl1200.76020
28. [28] Peetre J., A new approach in interpolation spaces, Studia Math., 1970, 34, 23–42 Zbl0188.43602
29. [29] Rao M.M., Ren Z.D., Theory of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math., 146, Marcel Dekker, New York, 1991 Zbl0724.46032
30. [30] Troianiello G.M., Elliptic Differential Equations and Obstacle Problems, Univ. Ser. Math., Plenum Press, New York, 1987 Zbl0655.35002
31. [31] Wolf J., Interior C 1,α-regularity of weak solutions to the equations of stationary motions of certain non-Newtonian fluids in two dimensions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 2007, 10(2), 317–340 

1. Václav Mácha, Partial regularity of solution to generalized Navier-Stokes problem
2. Cholmin Sin, Sin-Il Ri, Existence of weak solutions for steady flows of electrorheological fluid with Navier-slip type boundary conditions