Boundary regularity of flows under perfect slip boundary conditions

• Autorzy: Petr Kaplický , Jakub Tichý
• Języki publikacji: EN

Abstrakty

EN

We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions and we assume that the nonlinear elliptic operator satisfies non-standard ϕ-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity.

Słowa kluczowe

EN

Boundary regularity; Perfect Slip Boundary Condition; Generalized Stokes System

Kategorie tematyczne

• 76D03: Existence, uniqueness, and regularity theory
• 35J60: Nonlinear elliptic equations
• 35Q35: PDEs in connection with fluid mechanics
• 35B65: Smoothness and regularity of solutions

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 7
• Strony: 1243-1263

Daty

wydano

2013-07-01

online

2013-04-26

Twórcy

autor

Petr Kaplický

• Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic

autor

Jakub Tichý

• Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic

Bibliografia

• [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
• [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]
• [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]
• [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]
• [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] 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]
• [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]
• [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]
• [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]
• [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]
• [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] 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]
• [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]
• [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
• [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
• [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]
• [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]
• [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]
• [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]
• [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
• [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] 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]
• [23] Krasnosel’skiĭ M.A., Rutickiĭ Ja.B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961
• [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] 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
• [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
• [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
• [28] Peetre J., A new approach in interpolation spaces, Studia Math., 1970, 34, 23–42
• [29] Rao M.M., Ren Z.D., Theory of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math., 146, Marcel Dekker, New York, 1991
• [30] Troianiello G.M., Elliptic Differential Equations and Obstacle Problems, Univ. Ser. Math., Plenum Press, New York, 1987
• [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

Identyfikatory

DOI

10.2478/s11533-013-0232-x