  We prove the solvability of the parabolic $L^p$ Dirichlet boundary value
problem for $1 < p \leq \infty$ for a PDE of the form $u_t = \mbox{div} (A
\nabla u) + B \cdot \nabla u$ on time-varying domains where the coefficients
$A= [a_{ij}(X, t)]$ and $B=[b_i]$ satisfy a certain natural small Carleson
condition.
  This result brings the state of affairs in the parabolic setting up to the
elliptic standard.
  Furthermore, we establish that if the coefficients of the operator $A,\,B$
satisfy a vanishing Carleson condition and the time-varying domain is of VMO
type then the parabolic $L^p$ Dirichlet boundary value problem is solvable for
all $1 < p \leq \infty$.
  This result is related to results in papers by Maz\'ya, Mitrea and
Shaposhnikova, and Hofmann, Mitrea and Taylor where the fact that boundary of
domain has normal in VMO or near VMO implies invertibility of certain boundary
operators in $L^p$ for all $1 < p \leq \infty$ which then (using the method of
layer potentials) implies solvability of the $L^p$ boundary value problem in
the same range for certain elliptic PDEs.
  Our result does not use the method of layer potentials, since the
coefficients we consider are too rough to use this technique but remarkably we
recover $L^p$ solvability in the full range of $p$'s as the two papers
mentioned above.
