Advertisement

Journal of Mathematical Sciences

, Volume 232, Issue 3, pp 390–401 | Cite as

A Counterexample Related to the Regularity of the p-Stokes Problem

  • M. Křepela
  • M. Růžička
Article
  • 14 Downloads

We construct a solenoidal vector field u belonging to \( {W}^{2,q}\left(\Omega \right)\cap {W}_0^{1,s}\left(\Omega \right),q\in \left(1,n\right),s\in \left(1,\infty \right) \), such that (1 + |Du|)p − 2, p ∈ (1, ∞), p ≠ 2, does not belong to the Muckenhoupt class A(Ω). Thus, one cannot use the Korn inequality in weighted Lebesgue spaces to prove the natural regularity of the p-Stokes problem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Wiley, New York etc. (1987).Google Scholar
  2. 2.
    J. Málek, K. R. Rajagopal, and M. Růžička, “Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity,” Math. Models Methods Appl. Sci. 5, No. 6, 789–812 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Málek and K. R. Rajagopal, “Mathematical issues concerning the Navier-Stokes equations and some of its generalizations,” In: Evolutionary Equations. II, pp. 371–459, Elsevier, Amsterdam (2005),Google Scholar
  4. 4.
    E. Acerbi and N. Fusco, “Regularity for minimizers of nonquadratic functionals: the case 1 < p < 2,” J. Math. Anal. Appl. 140, No. 1, 115–135 (1989).Google Scholar
  5. 5.
    M. Giaquinta and G. Modica, “Remarks on the regularity of the minimizers of certain degenerate functionals,” Manuscripta Math. 57, No. 1, 55–99 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore (2003).Google Scholar
  7. 7.
    P. Kaplický, J. Málek, and J. Stará, “C 1-solutions to a class of nonlinear fluids in two dimensions – stationary Dirichlet problem,” J. Math. Sci., New York 109, No. 5, 1867-1892 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    C. Ebmeyer, “Steady flow of fluids with shear-dependent viscosity under mixed boundary value conditions in polyhedral domains,” Math. Models Methods Appl. Sci. 10, No. 5, 629–650 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    H. Beirão da Veiga, “On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions,” Commun. Pure Appl. Math. 58, No. 4, 552–577 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    C. Ebmeyer, “Regularity in Sobolev spaces of steady flows of fluids with shear-dependent viscosity,” Math. Methods Appl. Sci. 29, No. 14, 1687–1707 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    H. Beirão da Veiga, “On the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem,” J. Eur. Math. Soc. 11, No. 1, 127–167 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    F. Crispo and C. R. Grisanti, “On the existence, uniqueness and \( {C}^{1,\upgamma}\left(\overline{\Omega}\right)\cap {W}^{2,2}\left(\Omega \right) \) regularity for a class of shear-thinning fluids,” J. Math. Fluid Mech. 10, No. 4, 455–487 (2008).Google Scholar
  13. 13.
    H. Beirão da Veiga, “Turbulence models, p-fluid flows, and W 2,l-regularity of solutions,” Commun. Pure Appl. Anal. 8, No. 2, 769–783 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    F. Crispo, “A note on the global regularity of steady flows of generalized Newtonian fluids,” Port. Math. 66, No. 2, 211–223 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    H. Beirão da Veiga, “Navier-Stokes equations with shear thinning viscosity. Regularity up to the boundary,” J. Math. Fluid Mech. 11, No. 2, 258–273 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    H. Beirão da Veiga, “On the global regularity of shear thinning flows in smooth domains,” J. Math. Anal. Appl. 349, No. 2, 335–360 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    L. C. Berselli, “On the W 2,q-regularity of incompressible fluids with shear-dependent viscosities: the shear-thinning case,” J. Math. Fluid Mech. 11,, No. 2, 171–185 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    H. Beirão da Veiga, P. Kaplický, and M. Růžička, “Boundary regularity of shear thickening flows,” J. Math. Fluid Mech. 13, No. 3, 387–404 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    L. C. Berselli and M. Růžička, “Global regularity properties of steady shear thinning flows,” J. Math. Anal. Appl. 450, No. 2, 839–871 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    G. A. Seregin and T. N. Shilkin, “Regularity for minimizers of some variational problems in plasticity theory,” J. Math. Sci., New York 99, No. 1, 969–988 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    F. Crispo and P. Maremonti, “A high regularity result of solutions to modified p-Stokes equations,” Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 118, 97–129 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J. Nečas, “Sur le normes équivalentes dans \( {W}_p^k\left(\Omega \right) \) et sur la coercivité des formes formellement positives,” Sémin. Équ. Dériv. Partielles 317, 102–128 (1966).Google Scholar
  23. 23.
    L. Diening, M. Růžička, and K. Schumacher, “A decomposition technique for John domains,” Ann. Acad. Sci. Fenn., Math. 35, No. 1, 87–114 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Trans. Am. Math. Soc. 165, 207–226 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL (1992).Google Scholar
  26. 26.
    C. Amrouche and V. Girault, “Problèmes généralisés de Stokes,” Port. Math. 49, No. 4, 463–503 (1992).zbMATHGoogle Scholar
  27. 27.
    C. Bär, Elementary Differential Geometry, Cambridge Univ. Press, Cambridge (2010).Google Scholar
  28. 28.
    L. C. Berselli, L. Diening, and M. Růžička, “Existence of strong solutions for incompressible fluids with shear dependent viscosities,” J. Math. Fluid Mech. 12,, No. 1, 101–132 (2010).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied MathematicsUniversity of FreiburgFreiburgGermany

Personalised recommendations