  In this paper we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of
the parabolic operator $L_{t}=\partial_{t}-\Delta+b(t,x)\cdot\nabla$, where for
every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the
case that $b$ belongs to the Lebesgue space
$L^{l}\left(0,T;L^{q}\left(\mathbb{R}^{n}\right)\right)$. The regularity of
weak solutions to the parabolic equation $L_{t}u=0$ depends critically on the
value of the parabolic exponent $\gamma=\frac{2}{l}+\frac{n}{q}$. Without the
divergence-free condition on $b$, the regularity of weak solutions has been
established when $\gamma\leq1$, and the heat kernel estimate has been obtained
as well, except for the case that $l=\infty,q=n$. The regularity of weak
solutions was deemed not true for the critical case
$L^{\infty}\left(0,T;L^{n}\left(\mathbb{R}^{n}\right)\right)$ for a general
$b$, while it is true for the divergence-free case, and a written proof can be
deduced from the results in [Semenov, 2006]. One of the results obtained in the
present paper establishes the Aronson type estimate for critical and
supercritical cases and for vector fields $b$ which are divergence-free. We
will prove the best possible lower and upper bounds for the fundamental
solution one can derive under the current approach. The significance of the
divergence-free condition enters the study of parabolic equations rather
recently, mainly due to the discovery of the compensated compactness. The
interest for the study of such parabolic equations comes from its connections
with Leray's weak solutions of the Navier-Stokes equations and the Taylor
diffusion associated with a vector field where the heat operator $L_{t}$
appears naturally.
