1. CJM Online first
||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 CalderÃ³n-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)
||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
Keywords:initial $L^p$ Dirichlet problem, second order parabolic equations in divergence form, non-cylindrical domains, reverse HÃ¶lder inequalities
3. CJM 1997 (vol 49 pp. 798)