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2013 | 11 | 7 | 1243-1263
Tytuł artykułu

Boundary regularity of flows under perfect slip boundary conditions

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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.
Twórcy
  • Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic, kaplicky@karlin.mff.cuni.cz
autor
  • Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic, kuba.tichy@gmail.com
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]
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  • [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
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  • [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]
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  • [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
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Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0232-x
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