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

TL;DR

This paper establishes growth theorems and interior Harnack inequalities for second-order non-divergence parabolic equations with critical Morrey space drifts, extending classical estimates to unbounded drift scenarios.

Contribution

It introduces a variant of the Aleksandrov-Bakelman-Pucci-Krylov-Tso estimate for equations with drifts in Morrey spaces, advancing understanding of parabolic equations with unbounded coefficients.

Findings

Proved a new ABP-Krylov estimate with Morrey space drift

Derived growth theorems for solutions

Established interior Harnack inequality under critical Morrey space conditions

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}.