Parabolic equations with singular divergence-free drift vector fields

TL;DR

This paper proves the existence, uniqueness, and regularity of solutions for a class of parabolic equations with divergence-free, singular drift vector fields, extending classical results to non-symmetric elliptic operators with rough coefficients.

Contribution

It establishes fundamental solutions and Aronson estimates for parabolic operators with divergence-free drifts in BMO^{-1} space, a significant extension to non-symmetric elliptic operators.

Findings

Existence of fundamental solutions with Aronson estimates.

Uniqueness and regularity of solutions for initial data in L^2.

Applicability to non-symmetric elliptic operators in divergence form.

Abstract

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 $L=\nu\Delta+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;\xi,\tau)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension $n$, the elliptic constant $\nu$ and the norm $\left\Vert u\right\Vert _{L^{\infty}(\textrm{BMO}^{-1})}$. Therefore the existence and uniqueness of the parabolic equation $\left(L-\partial_{t}\right)v=0$ 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.