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

1612.06110

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### Abstract

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 $$\Gamma$$ -- .

### Author and article information

###### Journal
2016-12-19

math.AP
ccsd
 ScienceOpen disciplines: Analysis