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

Abstract

In this paper we construct a conservative Markov semi-group with generator $L=\Delta +b\cdot \nabla$ on ${\mathbb{R}}^n$, where $b$ is a divergence-free vector field which belongs to $L^2\cap L^p$ with $\frac{n}{2}<p$. 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.

12 pages

References in corpus (1)

• Conservativeness criteria for generalized Dirichlet forms