Parabolic equations with singular divergence-free drift vector fields

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\nu \mathrm{\Delta }+u(x,t)\cdot \mathrm{\nabla }$, where $u(\cdot ,t)$ is a time-dependent vector field in ${\mathbb{R}}^n$, which is divergence-free in distribution sense, i.e. $\mathrm{\nabla }\cdot u=0$. Suppose $u\in L_{t}^{\infty}($\text{BMO}$_{x}^{-1})$. We show the existence of the fundamental solution $\Gamma (x,t;\xi ,\tau )$ of the parabolic operator $L-{\mathrm{\partial }}_t$, and show that $\Gamma$ satisfies the Aronson estimate with a constant depending only on the dimension $n$, the elliptic constant $\nu$ and the norm ${\parallel u\parallel }_{L^{\mathrm{\infty }}({\text{BMO}}^{-1})}$. Therefore the existence and uniqueness of the parabolic equation $(L-{\mathrm{\partial }}_t)v=0$ are established for initial data in $L^2$-space, and their regularity is obtained too. In fact, we establish these results for a general non-symmetric elliptic operator in divergence form.