### 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.

Original language | English (US) |
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Pages (from-to) | 746-785 |

Number of pages | 40 |

Journal | Stochastics and Partial Differential Equations: Analysis and Computations |

Volume | 6 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2018 |

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### Keywords

- 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