Abstract
Regularity properties of solutions to the stationary generalized Stokes system are studied. The extra stress tensor is assumed to have a growth given by some N-function, which includes the situation of p-growth. We show results about differentiability of weak solutions. As a consequence we obtain the gradient L q estimates for the problem. These estimates are applied to the stationary generalized Navier Stokes equations.
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Acerbi E., Fusco N.: Regularity for minimizers of nonquadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140(1), 115–135 (1989)
Adams, R.A.: Sobolev spaces, Pure and Applied Mathematics. Academic Press, New York, vol. 65 (1975)
Belenki L., Diening L., Kreuzer Ch.: Optimality of an adaptive finite element method for the p-Laplacian equation. IMA J. Numer. Anal. 32(2), 484–510 (2011)
Breit, D.: Analysis of generalized Navier-Stokes equations for stationary shear thickening flow. Nonlinear Anal. (2012) doi:10.1016/j.na.2012.05.003
Breit D., Fuchs M.: The nonlinear Stokes problem with general potentials having superquadratic growth. J. Math. Fluid Mech. 13(3), 371–385 (2011)
Breit D., Stroffolini B., Verde A.: A general regularity theorem for functionals with \({\varphi}\) -growth. J. Math. Anal. sAppl. 383(1), 226–233 (2011)
Caffarelli L.A., Peral I.: On W 1, p estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51(1), 1–21 (1998)
Calderón A.P., Zygmund A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)
Campanato S., Stampacchia G.: Sulle maggiorazioni in L p nella teoria delle equazioni ellittiche. Boll. Un. Mat. Ital. (3) 20, 393–399 (1965)
Diening L., Ettwein F.: Fractional estimates for non-differentiable elliptic systems with general growth. Forum Mathematicum 20(3), 523–556 (2008)
Diening L., Růžička M.: Interpolation operators in Orlicz Sobolev spaces. Num. Math. 107(1), 107–129 (2007)
Diening L., Stroffolini B., Verde A.: Everywhere regularity of functionals with \({\varphi}\) -growth. Manuscripta Mathematica 129(4), 449–481 (2009)
Diening L., Kreuzer C.: Linear convergence of an adaptive finite element method for the p-Laplacian equation. SIAM J. Numer. Anal. 46(2), 614–638 (2008)
Diening L., Růžička M., Schumacher K.: A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Math. 35(1), 87–114 (2010)
Donaldson T.K., Trudinger N.S.: Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal. 8, 52–75 (1971)
Fuchs, M.: Stationary flows of shear thickening fluids in 2d. J. Math. Fluid Mech. (in press)
Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I: Linearized steady problems, Springer, New York (1994)
Giaquinta M.: Multiple Integrals in the Calculus of Variations and nonlinear elliptic Systems. Lectures in Mathematics. Ann. Math. Stud. Princeton University Press, Princeton (1982)
Habermann J.: Calderón-Zygmund estimates for higher order systems with p(x) growth. Math. Z. 258(2), 427–462 (2008)
Hopf E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)
Iwaniec T.: On L p-integrability in PDEs and quasiregular mappings for large exponents. Ann. Acad. Sci. Fenn. Ser. A I Math. 7(2), 301–322 (1982)
Iwaniec T.: Projections onto gradient fields and L p-estimates for degenerated elliptic operators. Studia Math. 75(3), 293–312 (1983)
Kaplický, P., Málek, J., Stará, J.: C 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. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 30, 89–121, 297
Kristensen J., Mingione G.: The singular set of minima of integral functionals, Arch. Arch. Ration. Mech. Anal. 180(3), 331–398 (2006)
Ladyženskaja O.A.: Modifications of the Navier-Stokes equations for large gradients of the velocities. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 126–154 (1968)
Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)
Lions, J.L.: Problèmes aux limites dans les équations aux dérivées partielles. Les Presses de l’Université de Montréal, Montreal (1965)
Mingione G.: Nonlinear aspects of Calderón-Zygmund theory. Jahresber. Dtsch. Math.-Ver. 112(3), 159–191 (2010)
Naumann J.: On the differentiability of weak solutions of a degenerate system of PDEs in fluid mechanics. Ann. Mat. Pura Appl. (4) 151, 225–238 (1988)
Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York (1991)
Verde A.: Calderón-Zygmund estimates for systems of \({\varphi}\) -growth. J. Convex Anal. 18, 67–84 (2011)
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. (8) 10(2), 317–340 (2007)
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Diening, L., Kaplický, P. L q theory for a generalized Stokes System. manuscripta math. 141, 333–361 (2013). https://doi.org/10.1007/s00229-012-0574-x
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DOI: https://doi.org/10.1007/s00229-012-0574-x