Initial-boundary value problems for continuity equations with BV coefficients

TL;DR

This paper proves well-posedness for initial-boundary value problems of continuity equations with BV coefficients without boundary orientation conditions, highlighting potential non-uniqueness when BV regularity deteriorates at the boundary.

Contribution

It establishes well-posedness results for continuity equations with BV coefficients without boundary orientation constraints and discusses boundary regularity effects on uniqueness.

Findings

Well-posedness is achieved for BV coefficient continuity equations.

Boundary orientation conditions are not necessary for well-posedness.

Loss of BV regularity at the boundary can lead to non-uniqueness.

Abstract

We establish well-posedness of initial-boundary value problems for continuity equations with BV (bounded total variation) coefficients. We do not prescribe any condition on the orientation of the coefficients at the boundary of the domain. We also discuss some examples showing that, regardless the orientation of the coefficients at the boundary, uniqueness may be violated as soon as the BV regularity deteriorates at the boundary.