Ergodicity and Hopf–Lax–Oleinik formula for fluid flows evolving around a black hole under a random forcing View Full Text


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Article Info

DATE

2018-12

AUTHORS

Yuri Bakhtin, Philippe G. LeFloch

ABSTRACT

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. More... »

PAGES

746-785

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Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s40072-018-0119-8

DOI

http://dx.doi.org/10.1007/s40072-018-0119-8

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1104379535


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