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

Abstract : This article studies the solutions in H 1 of a steady transport equation with a divergence-free driving velocity that is W 1,∞ , 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 Γ − 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 Γ − only in the case where the normal component of u does not vanish at the boundary of Γ −. In the case where the normal component of u vanishes at the boundary of Γ − , under appropriate assumptions, we construct local H 1 solutions in the neighborhood of the end-points of Γ − , which allow us to establish existence and uniqueness of the solution in H 1 for the transport equation with a Dirichlet boundary condition on Γ − .
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Contributor : Jean Marie Bernard <>
Submitted on : Saturday, December 17, 2016 - 7:11:37 PM
Last modification on : Friday, July 20, 2018 - 11:13:04 AM
Long-term archiving on : Tuesday, March 21, 2017 - 9:21:01 AM

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  • HAL Id : hal-01418930, version 1
  • ARXIV : 1612.06110

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Jean-Marie Bernard. Solutions H^1 of the steady transport equation in a bounded polygon with a fully non-homogeneous velocity. 2016. ⟨hal-01418930⟩

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