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Journal of Mathematical Sciences

, Volume 236, Issue 4, pp 399–412 | Cite as

LlogL-Integrability of the Velocity Gradient for Stokes System with Drifts in L(BMO1)

  • J. Burczak
  • G. Seregin
Article
  • 5 Downloads

For any weak solution of the Stokes system with drifts in L(BMO−1), a reverse Hölder inequality and LlogL-higher integrability of the velocity gradients are proved.

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References

  1. 1.
    A. Bonami, T. Iwaniec, P. Jones, and M. Zinsmeister, “On the product of functions in BMO and H 1,” Ann. Inst. Fourier, 57, No. 5, 1405–1439 (2007).MathSciNetCrossRefGoogle Scholar
  2. 2.
    H.-J. Choe and M. Yang, “Local kinetic energy and singularities of the incompressible Navier–Stokes equations,” arXiv:1705.04561 (2017).Google Scholar
  3. 3.
    L. Eskauriaza, G. A. Seregin, and V. Šverák, “L 3,∞-solutions of Navier–Stokes equations and backward uniqueness,” Uspekhi Mat. Nauk, 58, No. 2 (350), 3–44 (2003).Google Scholar
  4. 4.
    S. Friedlander and V. Vicol, “Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics,” Ann. Inst. H. Poincaré, 28, 2, 283–301 (2011).MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, New Jersey (1983).zbMATHGoogle Scholar
  6. 6.
    L. Greco, T. Iwaniec, and G. Moscariello, “Limits of the improved integrability of the volume forms,” Indiana Univ. Math. J., 44, No. 2, 305–339 (1995).MathSciNetCrossRefGoogle Scholar
  7. 7.
    T. Iwaniec and A. Verde, “On the operator L(f) = flog| f|,” J. Funct. Anal., 169, No. 2, 391–420 (1999).MathSciNetCrossRefGoogle Scholar
  8. 8.
    T. Iwaniec and J. Onninen, “H 1-estimates of Jacobians by subdeterminants,” Math. Ann., 324, No. 2, 341–358 (2002).MathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Koch, N. Nadirashvili, A. Seregin, and V. Šverák, “Liouville theorems for the Navier–Stokes equations and applications,” Acta Math., 203, No. 1, 83–105 (2009).MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. Koch and D. Tataru, “Well-posedness for the Navier–Stokes equations,” Adv. Math., 157, No. 1, 22–35 (2001).MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. Liskevich and Q. Zhang, “Extra regularity for parabolic equations with drift terms,” Manuscripta Math., 113, No. 2, 191–209 (2004).MathSciNetCrossRefGoogle Scholar
  12. 12.
    V. G. Maz’ya and I. E. Verbitsky, “Form boundedness of the general second-order differential operator,” Comm. Pure Appl. Math., 59, No. 9, 1286–1329 (2006).MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. I. Nazarov and N. N. Ural’tseva, “The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients,” St. Petersburg Math. J., 23, No. 1, 93–115 (2012).MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. E. Schonbek and G. Seregin, “Time decay for solutions to the Stokes equations with drift,” to appear in Commun. Contemp. Math.Google Scholar
  15. 15.
    G. A. Seregin, “Reverse Hölder inequality for a class of suitable weak solutions to the Navier-Stokes equations,” Zap. Nauchn. Semin. POMI, 362, 325–336 (2008).Google Scholar
  16. 16.
    G. Seregin, L. Silvestre, V. Šverák, and A. Zlatoš, “On divergence-free drifts,” J. Differential Equations, 252, 1, 505–540 (2012).MathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Seregin and V. Šverák, “On Type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations,” Comm. PDEs, 34, 171–201 (2009).MathSciNetCrossRefGoogle Scholar
  18. 18.
    L. Silvestre and V. Vicol, “Hölder continuity for a drift-diffusion equation with pressure,” Ann. Inst. H. Poincaré, 29, No. 4, 637–652 (2012).MathSciNetCrossRefGoogle Scholar
  19. 19.
    E. Stein, “Note on the class LlogL, Studia Math., 32, 305–310 (1969).MathSciNetCrossRefGoogle Scholar
  20. 20.
    E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey (1970).zbMATHGoogle Scholar
  21. 21.
    E. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey (1993).zbMATHGoogle Scholar
  22. 22.
    Q. Zhang, “Local estimates on two linear parabolic equations with singular coefficients,” Pacific J. Math., 223, No. 2, 367–396 (2006).MathSciNetCrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.Institute of MathematicsWarsawPoland
  2. 2.University of OxfordOxfordUK
  3. 3.St.Petersburg Department of Steklov Mathematical InstituteSt.PetersburgRussia

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