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In this paper, we show uniqueness and stability for the piecewise-smoothsolutions 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 Hilberttransform, a nonlocal operator. We show stability and uniqueness for solutionsamongst a larger class than the uniqueness result in Bressan and Zhang. Thesolutions we consider are measurable and bounded, satisfy at least one entropycondition, and verify a strong trace condition. We do not have smallnessassumptions. We use the relative entropy method and theory of shifts (seeVasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 --376, 2008]).

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