On the heat equation with singular drift

TL;DR

This paper establishes maximum modulus estimates for solutions to the heat equation with singular Morrey drift, applicable under specific integrability conditions, and extends to higher-order Laplacians.

Contribution

It provides new maximum modulus estimates for heat equations with Morrey drift, generalizing previous results and adaptable to higher-order Laplacian equations.

Findings

Maximum modulus estimates in terms of $L_{q,p}$-norms

Applicability to drifts satisfying Ladyzhenskaya-Prodi-Serrin condition

Technique adaptable to higher-order Laplacian equations

Abstract

We prove the maximum modulus estimates in terms of the $L_{q,p}$-norm of the free term for solutions of the heat equation with Morrey drift for any $q,p$ satisfying $d/p+2/q<2$ and any order of integration in the definition of the $L_{q,p}$-norm. An application to the case of $b$ satisfying the Ladyzhenskaya-Prodi-Serrin condition is given. The technique is easily adaptable to equations with Laplacians of order $\geq 1$.