On the heat equation with drift in $L_{d+1}$

TL;DR

This paper investigates the heat equation with drift in the Lebesgue space $L_{d+1}$, establishing unique solvability under certain conditions on the free term and function spaces involved.

Contribution

It proves the unique solvability of the heat equation with drift in a novel function class when the free term is in a high enough Lebesgue space.

Findings

Unique solvability in a new class of functions

Conditions on the free term in $L_q$ for solvability

Function space properties for solutions

Abstract

In this paper we deal with the heat equation with drift in $L_{d+1}$. Basically, we prove that, if the free term is in $L_{q}$ with high enough $q$, then the equation is uniquely solvable in a rather unusual class of functions such that $\partial_{t}u, D^{2}u\in L_{p}$ with $p<d+1$ and $Du\in L_{q}$.