SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2491-2530, January 2020.
In this paper, we show uniqueness and stability for the piecewise-smooth solutions, with a single discontinuity, to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15 (2017), pp. 165--184]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x={H}[u]$, where ${H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for these solutions among a larger class than the uniqueness result in Bressan and Zhang. We show stability and uniqueness for the Bressan--Zhang solutions in the large class of solutions which are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have smallness assumptions. We make no assumptions on the discontinuities in the solutions we consider. We use the relative entropy method and theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4 (2008), pp. 323--376]). Our results also apply to a more general class of nonlocal equations beyond the Burgers--Hilbert equation.

中文翻译：

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非局部标量守恒律分段光滑解的稳定性和唯一性及其在Burgers-Hilbert方程中的应用

2020年1月，SIAM数学分析期刊，第52卷，第3期，第2491-2530页。
在本文中，我们显示了由Bressan和Zhang [Commun。提出的Burgers-Hilbert方程具有单个不连续性的分段光滑解的唯一性和稳定性。数学。科学，15（2017），165--184]。Burgers--Hilbert方程为$ u_t +（\ frac {u ^ 2} {2}）_ x = {H} [u] $，其中$ {H} $是希尔伯特变换，它是一个非局部运算符。与Bressan和Zhang的唯一性结果相比，我们在更大的类别中显示了这些解决方案的稳定性和唯一性。我们在大量可测量且有界，至少满足一个熵条件并验证强跟踪条件的解决方案类别中显示了Bressan-Zhang解的稳定性和唯一性。我们没有小小的假设。我们对所考虑的解决方案中的不连续性不做任何假设。我们使用相对熵方法和位移理论（请参见Vasseur [《微分方程手册：进化方程》，第4版（2008），第323--376页）。我们的结果还适用于Burgers-Hilbert方程以外的更通用的非局部方程类。