A revisit on well-posedness of a boundary value problem of a stationary advection equation without the separation condition

TL;DR

This paper revisits the well-posedness of a stationary advection boundary value problem, offering new sufficient conditions that relax the traditional separation condition between inflow and outflow boundaries.

Contribution

It introduces alternative sufficient conditions for well-posedness in $L^p$ spaces, expanding the understanding beyond the classical separation condition.

Findings

Established new sufficient conditions for well-posedness.

Extended the range of $p$ for which the problem is well-posed.

Provided theoretical insights into boundary value problems of advection equations.

Abstract

We consider a boundary value problem of a stationary advection equation in a bounded domain with Lipschitz boundary. It is known to be well-posed in $L^p$-based function spaces for $1 < p < \infty$ under the separation condition of the inflow and the outflow boundaries. In this article, we provide another sufficient condition for the well-posedness with $1 \leq p \leq \infty$.