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

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]).

arXiv comment46 pages

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Cited by in corpus (6)

• Highest Cusped Waves for the Burgers-Hilbert equation
• Sharp a-contraction estimates for small extremal shocks
• Finite time stability for the Riemann problem with extremal shocks for a large class of hyperbolic systems
• $L^2$-stability $\&$ Minimal Entropy Conditions for Scalar Conservation Laws with Concave-Convex Fluxes
• Theory of shifts, shocks, and the intimate connections to $L^2$-type a posteriori error analysis of numerical schemes for hyperbolic problems
• Piecewise regular solutions to scalar balance laws with singular nonlocal sources