On elliptic and parabolic equations with critical and supercritical drift term

We discuss the properties of weak solutions to boundary and initial-boundary value problems with Dirichlet boundary conditions for elliptic and parabolic equations with non-divergence free drifts. Depending on the sign of the divergence of the drift, we will distinguish “non-spectral” and “spectral” cases. In each of these cases, we will study the issues of existence, uniqueness, regularity and stability of weak solutions. The talk is based on joint papers with M. Chernobai and M. Glazkov.