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

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 $\operatorname{div} b\le 0$.