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

Abstract

We show that the weak solutions of parabolic equation ${\partial }_{t}u-\Delta u+b(t,x)\cdot \nabla u=0$, $(t,x)\in (0,\infty )\times {\mathbb{R}}^d$, $d⩾3$, 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 $L^p$.

To appear is Osaka J. Math

References in corpus (2)

• On the best possible remaining term in the Hardy Inequality
• Gradient estimates for degenerate quasi-linear parabolic equations