Electron. J. Differential Equations, Vol. 2017 (2017), No. 75, pp. 1-22.

Local W^{1,p}-regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts

Tuoc Phan

We study weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted $L^p$-estimates for the gradients of weak solutions. Our results improve the classical one to the borderline case by replacing the $L^\infty$-assumption on solutions by solutions in the John-Nirenberg BMO space. The results are also generalized to parabolic equations in divergence form with small oscillation elliptic symmetric coefficients and therefore improve many known results.

Submitted December 31, 2016. Published March 20, 2017.
Math Subject Classifications: 35K10, 35K67, 35B45.
Key Words: Weighted Sobolev estimates; divergence-free drifts; Muckenhoupt weights; Hardy-Littlewood maximal functions.

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Tuoc Phan
Department of Mathematics
University of Tennessee, Knoxville
227 Ayress Hall, 1403 Circle Drive
Knoxville, TN 37996, USA
email: phan@math.utk.edu

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