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This article studies the solutions in H 1 of a steady transport equation witha divergence-free driving velocity that is W 1,$\infty$ , in a two-dimensionalbounded polygon. Since the velocity is assumed fully non-homogeneous on theboundary, existence and uniqueness of the solution require a boundary conditionon the open part $\Gamma$ -- where the normal component of u is strictlynegative. In a previous article, we studied the solutions in L 2 of this steadytransport equation. The methods, developed in this article, can be extended toprove existence and uniqueness of a solution in H 1 with Dirichlet boundarycondition on $\Gamma$ -- only in the case where the normal component of u doesnot vanish at the boundary of $\Gamma$ --. In the case where the normalcomponent of u vanishes at the boundary of $\Gamma$ -- , under appropriateassumptions, we construct local H 1 solutions in the neighborhood of theend-points of $\Gamma$ -- , which allow us to establish existence anduniqueness of the solution in H 1 for the transport equation with a Dirichletboundary condition on $\Gamma$ -- .

Solutions H^1 of the steady transport equation in a bounded polygon with a fully non-homogeneous velocity