Stoch PDE: Anal Comp https://doi.org/10.1007/s40072-018-0119-8 Ergodicity and Hopf–Lax–Oleinik formula for ﬂuid ﬂows evolving around a black hole under a random forcing 1 2 Yuri Bakhtin · Philippe G. LeFloch Received: 14 February 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We study the ergodicity properties of weak solutions to a relativistic gen- eralization 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 ran- dom 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’
Stochastical Partial Differential Equations – Springer Journals
Published: Jun 4, 2018
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