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Stokes problem; generalized Newtonian fluids; regularity; nonautonomous functionals; local minimizer
We discuss regularity results concerning local minimizers $u: \mathbb R^n\supset \Omega\rightarrow\mathbb R^n$ of variational integrals like \begin{align*} \int_{\Omega}\{F(\cdot ,\varepsilon (w))-f\cdot w\}\,dx \end{align*} defined on energy classes of solenoidal fields. For the potential $F$ we assume a $(p,q)$-elliptic growth condition. In the situation without $x$-dependence it is known that minimizers are of class $C^{1,\alpha }$ on an open subset $\Omega_{0}$ of $\Omega$ with full measure if $q< p\,\frac{n+2}{n}$ (for $n=2$ we have $\Omega_{0}=\Omega$). In this article we extend this to the case of nonautonomous integrands. Of course our result extends to weak solutions of the corresponding nonlinear Stokes type system.
[AM] Acerbi E., Mingione G.: Regularity results for stationary electro-rheological fluids. Arch. Rat. Mech. Anal 164 (2002), 213–259. DOI 10.1007/s00205-002-0208-7 | MR 1930392 | Zbl 1038.76058
[Br1] Breit D.: Regularitätssätze für Variationsprobleme mit anisotropen Wachstumsbedingungen. PhD thesis, Saarland University, 2009.
[Br2] Breit D.: New regularity theorems for non-autonomous variational integrals with $(p,q)$-growth. Calc. Var. 44 (2012), 101–129. DOI 10.1007/s00526-011-0428-5 | MR 2898773 | Zbl 1252.49060
[BF1] Bildhauer M., Fuchs M.: Variants of the Stokes problem: the case of anisotropic potentials. J. Math. Fluid Mech. 5 (2003), 364–402. DOI 10.1007/s00021-003-0072-8 | MR 2004292 | Zbl 1072.76019
[BF2] Bildhauer M., Fuchs M.: $C^{1,\alpha}$-solutions to non-autonomous anisotropic variational problems. Calc. Var. 24 (2005), no. 3, 309–340. DOI 10.1007/s00526-005-0327-8 | MR 2174429 | Zbl 1101.49029
[BF3] Bildhauer M., Fuchs M.: Partial regularity for variational integrals with $(s,\mu,q)$-growth. Calc. Var. 13 (2001), 537–560. DOI 10.1007/s005260100090 | MR 1867941 | Zbl 1018.49026
[BF5] Bildhauer M., Fuchs M.: A regularity result for stationary electrorheological fluids in two dimensions. Math. Methods Appl. Sci. 27 (2004), no. 13, 1607–1617. DOI 10.1002/mma.527 | MR 2077446 | Zbl 1058.76073
[BFZ] Bildhauer M., Fuchs M., Zhong X.: A lemma on the higher integrability of functions with applications to the regularity theory of two-dimensional generalized Newtonian fluids. Manuscripta Math. 116 (2005), no. 2, 135–156. DOI 10.1007/s00229-004-0523-4 | MR 2122416 | Zbl 1116.49018
[CGM] Cupini G., Guidorzi M., Mascolo E.: Regularity of minimizers of vectorial integrals with $p-q$ growth. Nonlinear Anal. 54 (2003), no. 4, 591-616. DOI 10.1016/S0362-546X(03)00087-7 | MR 1983438 | Zbl 1027.49032
[DER] Diening L., Ettwein F., Růžička M.: $C^{1,\alpha}$-regularity for electrorheological fluids in two dimensions. Nonlinear Differential Equations Appl. 14 (2007), no. 1–2, 207–217. DOI 10.1007/s00030-007-5026-z | MR 2346460 | Zbl 1132.76301
[ELM1] Esposito L., Leonetti F., G. Mingione G.: Sharp regularity for functionals with $(p,q)$-growth. J. Differential Equations 204 (2004), 5–55. DOI 10.1016/S0022-0396(04)00208-6 | MR 2076158 | Zbl 1072.49024
[ELM2] Esposito L., Leonetti F., Mingione G.: Regularity for minimizers of irregular integrals with $(p,q)$-growth. Forum Mathematicum 14 (2002), 245–272. DOI 10.1515/form.2002.011 | MR 1880913
[Fu] Fuchs M.: On quasistatic non-Newtonian fluids with power law. Math. Methods Appl. Sci. 19 (1996), 1225–1232. DOI 10.1002/(SICI)1099-1476(199610)19:15<1225::AID-MMA827>3.0.CO;2-U | MR 1410207
[FuGR] Fuchs M., Grotowski J., Reuling J.: On variational models for quasistatic Bingham fluids. Math. Methods Appl. Sci. 19 (1996), 991–1015. DOI 10.1002/(SICI)1099-1476(199608)19:12<991::AID-MMA810>3.0.CO;2-R | MR 1402153
[FuS] Fuchs M., Seregin G.: Variational methods for fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening. Math. Methods Appl. Sci. 22 (1999), 317–351. DOI 10.1002/(SICI)1099-1476(19990310)22:4<317::AID-MMA43>3.0.CO;2-A | MR 1671448 | Zbl 0928.76087
[Gi] Giaquinta M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Birkhäuser, Basel-Boston-Berlin, 1993. MR 1239172 | Zbl 0786.35001
[KMS] Kaplický P., Málek J., Stará J.: $C^{1,\alpha}$-solutions to a class of nonlinear fluids in two dimensions --- stationary Dirichlet problem. Zap. Nauchn. Sem. POMI 259 (1999), 122–144. Zbl 0978.35046
[La] Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York-London-Paris, 1969. MR 0254401 | Zbl 0184.52603
[MNRR] Málek J., Nečas J., Rokyta M., Růžička M.: Weak and Measure Valued Solutions to Evolutionary PDEs. Chapman & Hall, London-Weinheim-New York, 1996. MR 1409366 | Zbl 0851.35002
[Mo] Morrey C.B.: Multiple integrals in the calculus of variations. Grundlehren der math. Wiss. in Einzeldarstellungen, 130, Springer, Berlin-Heidelberg, 1966. MR 2492985 | Zbl 1213.49002
[NW] Naumann J., Wolf J.: Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids. J. Math. Fluid Mech. 7 (2005), 298–313. DOI 10.1007/s00021-004-0120-z | MR 2177130 | Zbl 1070.35023
[Ru] Růžička M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, 1748, Springer, Berlin, 2000. DOI 10.1007/BFb0104030 | MR 1810360 | Zbl 0968.76531
[Wo] Wolf J.: Interior $C^{1,\alpha}$-regularity of weak solutions to the equations of stationary motions of certain non-Newtonian fluids in two dimensions. Boll. Unione Mat. Ital. Sez. B (8) 10 (2007), 317–340. MR 2339444
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