### 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) |
---|---|

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 |

### Fingerprint

### 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

### Cite this

**Ergodicity and Hopf–Lax–Oleinik formula for fluid flows evolving around a black hole under a random forcing.** / Bakhtin, Yuri; LeFloch, Philippe G.

Research output: Contribution to journal › Article

}

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.

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 -