# Stability and uniqueness for piecewise smooth solutions to Burgers-Hilbert among a large class of solutions

## TL;DR

This paper proves the stability and uniqueness of piecewise-smooth solutions to the Burgers-Hilbert equation within a broad class of solutions, using the relative entropy method and shift theory, without smallness constraints.

## Contribution

It extends the uniqueness and stability results for the Burgers-Hilbert equation to a larger class of solutions, including measurable and bounded functions satisfying entropy and trace conditions.

## Key findings

- Proves stability and uniqueness for a broad class of solutions.
- Uses the relative entropy method and shift theory.
- No smallness assumptions are needed.

## Abstract

In this paper, we show uniqueness and stability for the piecewise-smooth solutions to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15(1):165--184, 2017]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x=\mathbf{H}[u]$ where $\mathbf{H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for solutions amongst a larger class than the uniqueness result in Bressan and Zhang. The solutions we consider are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have smallness assumptions. We use the relative entropy method and theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]).