We study the ergodicity properties of weak solutions to a relativistic generalization of Burgers equation posed on a curved background and, specifically, a Schwarzschild black hole. We investigate the interplay between the dynamics of shocks, a curved geometric background, and a random boundary forcing, and solve three problems of independent interest. First of all, we consider the standard Burgers equation on a half-line and establish a ‘one-force-one-solution’ principle when the random forcing at the boundary is sufficiently “strong” in comparison with the velocity of the solutions at infinity. Secondly, we consider the Burgers–Schwarzschild model and establish a generalization of the Hopf–Lax–Oleinik formula. This novel formula takes the curved geometry into account and allows us to establish the existence of bounded variation solutions. Thirdly, under a random boundary forcing in the vicinity of the horizon of the black hole, we prove the existence of a random global attractor and we again validate the ‘one-force-one-solution’ principle. Finally, we extend our main results to the pressureless Euler system which includes a transport equation satisfied by the integrated fluid density.

中文翻译：

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随机强迫作用下绕黑洞演化的流体流动的遍历性和Hopf–Lax–Oleinik公式

我们研究了弯曲背景上，尤其是Schwarzschild黑洞所构成的Burgers方程的相对论性广义化的弱解的遍历性质。我们研究了冲击动力学，弯曲的几何背景和随机边界强迫之间的相互作用，并解决了三个独立利益问题。首先，我们在半线上考虑标准的Burgers方程，并在边界处的随机强迫与无穷大处的速度相比足够“强”时，建立“一力一解”原理。 。其次，我们考虑了Burgers-Schwarzschild模型，并建立了Hopf-Lax-Oleinik公式的推广。这个新颖的公式考虑了弯曲的几何形状，并允许我们建立有界变化解的存在。第三，在黑洞视界附近的随机边界强迫下，我们证明了随机全局吸引子的存在，并且再次验证了“一力一解”原理。最后，我们将主要结果扩展到无压欧拉系统，该系统包括一个由积分流体密度满足的输运方程。