  This article studies the solutions in H 1 of a steady transport equation with
a divergence-free driving velocity that is W 1,$\infty$ , in a two-dimensional
bounded polygon. Since the velocity is assumed fully non-homogeneous on the
boundary, existence and uniqueness of the solution require a boundary condition
on the open part $\Gamma$ -- where the normal component of u is strictly
negative. In a previous article, we studied the solutions in L 2 of this steady
transport equation. The methods, developed in this article, can be extended to
prove existence and uniqueness of a solution in H 1 with Dirichlet boundary
condition on $\Gamma$ -- only in the case where the normal component of u does
not vanish at the boundary of $\Gamma$ --. In the case where the normal
component of u vanishes at the boundary of $\Gamma$ -- , under appropriate
assumptions, we construct local H 1 solutions in the neighborhood of the
end-points of $\Gamma$ -- , which allow us to establish existence and
uniqueness of the solution in H 1 for the transport equation with a Dirichlet
boundary condition on $\Gamma$ -- .
