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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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).
H.-J. Choe and M. Yang, “Local kinetic energy and singularities of the incompressible Navier–Stokes equations,” arXiv:1705.04561 (2017).
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).
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).
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, New Jersey (1983).
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).
T. Iwaniec and A. Verde, “On the operator L(f) = flog| f|,” J. Funct. Anal., 169, No. 2, 391–420 (1999).
T. Iwaniec and J. Onninen, “H 1-estimates of Jacobians by subdeterminants,” Math. Ann., 324, No. 2, 341–358 (2002).
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).
H. Koch and D. Tataru, “Well-posedness for the Navier–Stokes equations,” Adv. Math., 157, No. 1, 22–35 (2001).
V. Liskevich and Q. Zhang, “Extra regularity for parabolic equations with drift terms,” Manuscripta Math., 113, No. 2, 191–209 (2004).
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).
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).
M. E. Schonbek and G. Seregin, “Time decay for solutions to the Stokes equations with drift,” to appear in Commun. Contemp. Math.
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).
G. Seregin, L. Silvestre, V. Šverák, and A. Zlatoš, “On divergence-free drifts,” J. Differential Equations, 252, 1, 505–540 (2012).
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).
L. Silvestre and V. Vicol, “Hölder continuity for a drift-diffusion equation with pressure,” Ann. Inst. H. Poincaré, 29, No. 4, 637–652 (2012).
E. Stein, “Note on the class LlogL, Studia Math., 32, 305–310 (1969).
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey (1970).
E. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey (1993).
Q. Zhang, “Local estimates on two linear parabolic equations with singular coefficients,” Pacific J. Math., 223, No. 2, 367–396 (2006).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 459, 2017, pp. 35–57.
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Burczak, J., Seregin, G. LlogL-Integrability of the Velocity Gradient for Stokes System with Drifts in L∞(BMO−1). J Math Sci 236, 399–412 (2019). https://doi.org/10.1007/s10958-018-4120-6
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DOI: https://doi.org/10.1007/s10958-018-4120-6