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Search: All articles in the CJM digital archive with keyword Parabolic equation

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1. CJM Online first

Phan, Tuoc
Lorentz estimates for weak solutions of quasi-linear parabolic equations with singular divergence-free drifts
This paper investigates regularity in Lorentz spaces of weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions which are measurable in $(x,t)$-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John Nirenberg space. The results are even new when the drifts are identically zero because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón-Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known, available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a "double-scaling parameter" technique, and the maximal function free approach introduced by Acerbi and Mingione.

Keywords:gradient estimate, quasi-linear parabolic equation, divergence-free drift
Categories:35B45, 35K57, 35K59, 35K61

2. CJM 2013 (vol 66 pp. 429)

Rivera-Noriega, Jorge
Perturbation and Solvability of Initial $L^p$ Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains
For parabolic linear operators $L$ of second order in divergence form, we prove that the solvability of initial $L^p$ Dirichlet problems for the whole range $1\lt p\lt \infty$ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of $L$ satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of $p\gt 1$, the initial $L^p$ Dirichlet problem associated to $Lu=0$ over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

Keywords:initial $L^p$ Dirichlet problem, second order parabolic equations in divergence form, non-cylindrical domains, reverse Hölder inequalities
Category:35K20

3. CJM 1997 (vol 49 pp. 798)

Yu, Minqi; Lian, Xiting
Boundedness of solutions of parabolic equations with anisotropic growth conditions
In this paper, we consider the parabolic equation with anisotropic growth conditions, and obtain some criteria on boundedness of solutions, which generalize the corresponding results for the isotropic case.

Keywords:Parabolic equation, anisotropic growth conditions, generalized, solution, boundness
Categories:35K57, 35K99.

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