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Non-divergence Parabolic Equations of Second Order with Critical Drift in Morrey Spaces (1602.00819v1)

Published 2 Feb 2016 in math.AP

Abstract: We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift [-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)] in some domain $\Omega\subset \mathbb{R}{n+1}$. We prove a variant of Aleksandrov-Bakelman-Pucci-Krylov-Tso estimate with $L{p}$ norm of the inhomogeneous term for some number $p<n+1$. Based on it, we derive the growth theorems and the interior Harnack inequality. In this paper, we will only assume the drift $b$ is in certain Morrey spaces defined below which are critical under the parabolic scaling but not necessarily to be bounded. This is a continuation of the work in \cite{GC}.

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