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 \(\Gamma\) -- .