Markov semi-groups generated by elliptic operators with divergence-free drift

Abstract

In this paper we construct a conservative Markov semi-group with generator L=Δ+b⋅∇ on Rn, where b is a divergence-free vector field which belongs to L2∩Lp with n2<p. The research is motivated by the question of understanding the blow-up solutions of the fluid dynamic equations, which attracts a lot of attention in recent years.