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

In this paper we construct a conservative Markov semi-group with generator L =Δ +b ·∇on Rn, where bis a divergence-free vector field which belongs to L2∩Lpwith 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.