# Well-posedness for the continuity equation for vector fields with suitable modulus of continuity

## TL;DR

This paper establishes the well-posedness of scalar conservation laws with vector fields having specific continuity properties, demonstrating uniqueness of solutions even when divergence is unbounded, expanding the understanding of such PDEs.

## Contribution

It proves well-posedness under minimal assumptions on the velocity field's growth and continuity, without requiring divergence control, and applies this to cases with unbounded divergence.

## Key findings

- Proves well-posedness for scalar conservation laws with certain vector fields.
- Shows uniqueness of solutions in the atomic Hardy space H1.
- Handles vector fields with divergence as an unbounded BMO function.

## Abstract

We prove well-posedness of linear scalar conservation laws using only assumptions on the growth and the modulus of continuity of the velocity field, but not on its divergence. As an application, we obtain uniqueness of solutions in the atomic Hardy space, H1, for the scalar conservation law induced by a class of vector fields whose divergence is an unbounded BMO function.