Parabolic equations with singular divergence‐free drift vector fields

In this paper, we study an elliptic operator in divergence form but not necessarily symmetric. In particular, our results can be applied to elliptic operator L=νΔ+u(x,t)·∇, where u(·,t) is a time‐dependent vector field in ℝn, which is divergence‐free in the distributional sense, that is ∇·u=0. Suppose u∈L∞(0,∞;BMO−1(ℝn)). We show the existence of the fundamental solution Γ(x,t;ξ,τ) of the parabolic operator L−∂t and show that Γ satisfies the Aronson estimate with a constant depending only on the dimension n, the elliptic constant λ and the norm ∥u∥L∞t(BMO−1x). Therefore, the existence and uniqueness of solutions to the parabolic equation (L−∂t)v=0 are established for initial data in L2‐space, and their regularity theory is obtained too. In fact, we establish these results for a general non‐symmetric elliptic operator in divergence form.