STEADY TRANSPORT EQUATION IN THE CASE WHERE THE NORMAL COMPONENT OF THE VELOCITY DOES NOT VANISH ON THE BOUNDARY

This article studies the solutions in L-2 of a steady transport equation with a divergence-free driving velocity that is H-1 in a Lipschitz domain of R-d. Since the velocity is assumed fully nonhomogeneous on the boundary, existence and uniqueness of solution require a boundary condition. A new Green's formula allows us to define the normal component of zu on the boundary, where z denotes the stress and u the velocity. A substantial part of the article is devoted to properties of a truncature operator in the space where z and u. del z are L-2. By means of these properties, which allow us to prove density results, and by using in addition a nonbounded linear operator from L-2 to L-2, we establish existence and uniqueness of the solution for the transport equation with a boundary condition on the open part where the normal component of u is strictly negative.