On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data

TL;DR

This paper extends the theory of transport equations with non-smooth velocity fields to include inhomogeneous Dirichlet boundary conditions, establishing existence and uniqueness of solutions under less regular assumptions.

Contribution

It introduces a new notion of solutions for transport equations with boundary data and proves existence and uniqueness results, extending DiPerna-Lions theory to bounded domains with boundary conditions.

Findings

Existence of unique renormalized weak solutions for specified initial and boundary data.

Extension of DiPerna-Lions theory to bounded, possibly unbounded, domains with boundary conditions.

Mollification technique tailored to boundary normal direction to approximate solutions.

Abstract

A transport equation with a non-smooth velocity field is considered under inhomogeneous Dirichlet boundary conditions. The spatial gradient of the velocity field is assumed in $L^{p'}$ in space and the divergence of the velocity field is assumed to be bounded. By introducing a suitable notion of solutions, it is shown that there exists a unique renormalized weak solution for $L^p$ initial and boundary data for $1/p+1/p'=1$. Our theory is considered as a natural extension of the theory due to DiPerna and Lions (1989), where there is no boundary. Although a smooth domain is considered, it is allowed to be unbounded. A key step is a mollification of a solution. In our theory, mollification in the direction normal to the boundary is tailored to approximate the boundary data.