On the $L_1$--stability for parabolic equations with a supercritical drift term

TL;DR

This paper establishes the $L_1$-stability of weak solutions to certain parabolic equations with supercritical drift terms, under specific divergence conditions, contributing to the understanding of solution behavior under drift perturbations.

Contribution

It proves $L_1$-stability for parabolic equations with supercritical drift in $L_2$, under the divergence condition $ ext{div} b \u2264 0$, which is a novel stability result.

Findings

Proves existence and uniqueness of weak solutions.

Demonstrates $L_1$-stability under drift perturbations.

Identifies conditions on the drift for stability.

Abstract

In this paper we investigate the existence, uniqueness and stability of weak solutions of the initial boundary value problem with the Dirichlet boundary conditions for a parabolic equation with a drift $b\in L_2$. We prove $L_1$-stability of solutions with respect to perturbations of the drift $b$ in $L_2$ in the case if the drift satisfies the ``non-spectral'' condition $\operatorname{div} b\le 0$.