Feller evolution families and parabolic equations with form-bounded vector fields

Abstract

We show that the weak solutions of parabolic equation \(\partial_t u - \Delta u + b(t,x) \cdot \nabla u=0\), \((t,x) \in (0,\infty) \times \mathbb R^d\), \(d \geqslant 3\), for \(b(t,x)\) in a wide class of time-dependent vector fields capturing critical order singularities, constitute a Feller evolution family and, thus, determine a Feller process. Our proof uses an a priori estimate on the \(L^p\)-norm of the gradient of solution in terms of the \(L^q\)-norm of the gradient of initial function, and an iterative procedure that moves the problem of convergence in \(L^\infty\) to \(L^p\).