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

TL;DR

This paper investigates the existence and uniqueness of H^1 solutions to a steady transport equation with non-homogeneous boundary conditions in a polygon, extending previous L^2 results and addressing boundary behavior of the velocity.

Contribution

It extends previous work by establishing H^1 solutions for the steady transport equation with non-homogeneous boundary conditions, including cases where the velocity's normal component vanishes at boundary endpoints.

Findings

Proves existence and uniqueness of H^1 solutions under certain boundary conditions.

Develops methods to handle boundary points where the velocity normal component vanishes.

Constructs local solutions near boundary endpoints to ensure global H^1 solutions.

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 $\Gamma$ -- .