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

Yuri Bakhtin, Philippe G. LeFloch

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)746-785
Number of pages40
JournalStochastics and Partial Differential Equations: Analysis and Computations
Issue number4
StatePublished - Dec 1 2018


  • Burgers equation
  • Ergodicity
  • Hopf–Lax–Oleinik formula
  • One-force-one-solution principle
  • Random forcing
  • Schwarzschild black hole

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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