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

## TL;DR

This paper constructs a conservative Markov semi-group generated by an elliptic operator with divergence-free drift in Euclidean space, contributing to the understanding of fluid dynamic equations and their blow-up solutions.

## Contribution

It introduces a new class of Markov semi-groups with divergence-free drifts in specific Lp spaces, advancing the mathematical framework for fluid dynamics analysis.

## Key findings

- Constructed a Markov semi-group with divergence-free drift in L^2 ∩ L^p
- Provided insights into blow-up solutions of fluid dynamic equations
- Extended the theory of elliptic operators with divergence-free coefficients

## 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.