Non-divergence Parabolic Equations of Second Order with Critical Drift in Morrey Spaces

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}.

arXiv comment23 pages. arXiv admin note: substantial text overlap with arXiv:1511.01215