TY - JOUR

T1 - Ergodicity and Hopf–Lax–Oleinik formula for fluid flows evolving around a black hole under a random forcing

AU - Bakhtin, Yuri

AU - LeFloch, Philippe G.

N1 - Funding Information:
Acknowledgements The work of YB was partially supported by the National Science Foundation (NSF) through Grant DMS-1460595. This work was done when PLF was a visiting researcher at the Courant Institute of Mathematical Sciences (NYU) and was also partially supported by the Innovative Training Networks (ITN) Grant 642768 (ModCompShock).
Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Burgers equation

KW - Ergodicity

KW - Hopf–Lax–Oleinik formula

KW - One-force-one-solution principle

KW - Random forcing

KW - Schwarzschild black hole

UR - http://www.scopus.com/inward/record.url?scp=85061711461&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061711461&partnerID=8YFLogxK

U2 - 10.1007/s40072-018-0119-8

DO - 10.1007/s40072-018-0119-8

M3 - Article

AN - SCOPUS:85061711461

VL - 6

SP - 746

EP - 785

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

SN - 2194-0401

IS - 4

ER -