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Manuscripta Mathematica

, Volume 144, Issue 1–2, pp 51–90 | Cite as

Almost everywhere Hölder continuity of gradients to non-diagonal parabolic systems

  • Jan Burczak
Open Access
Article

Abstract

We present a local almost everywhere C 1,α -regularity result for a general class of p-nonlinear non-diagonal parabolic systems. The main part of the considered systems depends on space-time variable, solution and symmetric part of the gradient of solution. To obtain our result, we adapt for the symmetric-gradient case techniques developed for the full-gradient case by Duzaar, Mingione and coauthors.

Mathematics Subject Classification (2000)

MSC 35K55 MSC 35B65 MSC 35K92 

References

  1. 1.
    Apushkinskaya D., Bildhauer M., Fuchs M.: Steady states of anisotropic generalized Newtonian fluids. J. Math. Fluid Mech. 7(2), 261–297 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Beirão da Veiga H., Crispo F.: On the global W 2,q regularity for nonlinear N-systems of the p-Laplacian type in n space variables. Nonlinear Anal. 75(11), 4346–4354 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bojarski, B., Hajł asz, P.: Pointwise inequalities for Sobolev functions and some applications. Studia Math., 106 (1) (1993)Google Scholar
  4. 4.
    Breit D., Fuchs M.: The nonlinear Stokes problem with general potentials having superquadratic growth. J. Math. Fluid Mech. 13(3), 371–385 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Campanato S.: Equazioni paraboliche del secondo ordine e spazi \({\mathcal{L}^{2,\theta}(\Omega,\delta)}\). Ann. Mat. Pura Appl. 73(4), 55–102 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Desvilettes L., Villani C.: On a variant of Korn’s inequality arising in statistical medanics. ESAIM Control Optim. Calc. Var. 8, 603–619 (2002)Google Scholar
  7. 7.
    DiBenedetto E.: Degenerate parabolic systems. Springer, Berlin (1993)CrossRefGoogle Scholar
  8. 8.
    Diening L., Stroffolini B., Verde A.: The ϕ-harmonic approximation and the regularity of ϕ–harmonic maps. J. Differ. Equ. 253(7), 1943–1958 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Duzaar F., Mingione G.: Second order parabolic systems, optimal regularity and singular sets of solutions. AIHP – AN 22, 705–751 (2005)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Duzaar, F., Mingione, G., Steffen, K.: Parabolic Systems with Polynomial Growth and Regularity, Memoirs A.M.S. 214, 2011Google Scholar
  11. 11.
    Eidelman, S., Zhitarashu, N.: Parabolic Boundary Value Problems, Birkhäuser, 1998Google Scholar
  12. 12.
    Friedrichs K.: On the boundary-value problems of the theory of elasticity and Korn’s inequality. Ann. of Math. (2) 48, 441–471 (1947)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kaplický P., Málek J., Stará J.: Global-in-time Hlder continuity of the velocity gradients for fluids with shear-dependent viscosities. NoDEA Nonlinear Differential Equations Appl. 9(2), 175–195 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Korn A.: Die Eigenschwingungen eines elastischen Körpers mit ruhender Oberfläche. Akad. der Wiss., München, Math. phys. Klasse, Ber. 36, 351 (1906)Google Scholar
  15. 15.
    Kronz M.: Partial regularity results for minimizers of quasiconvex functionals of higher order. AIHP – AN 19, 81–112 (2002)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Nečas, J., Málek, J., Ruzička, M., Rokyta, M.: Weak and measure-valued solutions to evolutionary PDEs. Chapman & Hall (1996)Google Scholar
  17. 17.
    Prüss J., Bothe D.: Lp-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39(2), 379–421 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Schlag W.: Schauder and Lp estimates for parabolic systems via Campanato spaces. Comm. P.D.E. 21(7–8), 1141–1175 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Seregin G., Ladyzhenskaya O.: On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 1(4), 356–387 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Simon, L.: Theorems on regularity and singularity of energy minimizing maps, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996Google Scholar
  21. 21.
    Solonnikov V.: L p estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain. J. Math. Sci. 105(5), 2448–2484 (2001)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Tolksdorf P.: Everywhere-regularity for some quasilinear systems with a lack of ellipticity. Ann. Mat. Pura Appl. (4) 134, 241–266 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Uhlenbeck K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138(3-4), 219–240 (1977)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Institute of Mathematics, Polish Academy of SciencesWarsawPoland

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