We establish $L^p$, $2\le p\le\infty$ solvability of the Dirichlet boundary value problem for a parabolic equation $u_t-\mbox{div}(A\nabla u)- {B}\cdot\nabla u =0$ on time-varying domains with coefficient matrices $A=[a_{ij}]$ and ${B} =[b_{i}]$ that satisfy a small Carleson condition. The result is motivated by similar results for the elliptic equation
$\mbox{div}(A\nabla u) + {B}\cdot \nabla u =0$ that were established in
the papers Kenig, Pipher, Petermichl and myself. The result complements
the papers of Hofmann and Rivera-Noriega where solvability of parabolic
$L^p$ (for some large $p$) Dirichlet boundary value problem for
coefficients that satisfy large Carleson condition was established.

The main result says that given $p\in (2,\infty)$ there exists $C(p)>0$
such that if the Carleson norm of coefficients is small then the
Dirichlet boundary value problem for a parabolic equation is solvable in
$L^p$. This is a join work with S. Hwang.

I shall also discuss a second result (with J. Pipher and S. Petermich)
that shows that $C(p)\to \infty$ as $p\to\infty$ which is a quantitative
result complementing the previous results on $L^p$ solvability under
large Carleson condition (where only existence of $p<\infty$ is
established, but not how it might depend on the Carleson norm).