Weak transport equation on a bounded domain: stability theory apr\'es DiPerna-Lions

TL;DR

This paper investigates the stability of the weak transport equation for incompressible fluids in bounded domains, establishing existence, uniqueness, and stability results under zero boundary velocity conditions.

Contribution

It extends DiPerna-Lions stability theory to bounded domains for the weak transport equation with zero boundary velocity.

Findings

Proved existence and uniqueness of solutions.

Established stability results in bounded domains.

Analyzed the impact of boundary conditions on solution behavior.

Abstract

The article studies the transport equation that governes the motion of a fluid in a bounded domain, under the hypothesis of zero velocity at the boundary and supposing the incompressible nature of the fluid. Together with existence and uniqueness results, we study the DiPerna-Lions stability problem, in the case of a bounded domain.