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 journalArticle

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 languageEnglish (US)
Pages (from-to)746-785
Number of pages40
JournalStochastics and Partial Differential Equations: Analysis and Computations
Volume6
Issue number4
DOIs
StatePublished - Dec 1 2018

Fingerprint

Ergodicity
Forcing
Black Holes
Fluid Flow
Flow of fluids
Burgers Equation
Random Attractor
Euler System
Bounded variation
Global Attractor
Transport Equation
Weak Solution
Half line
Horizon
Shock
Infinity
Fluid
Fluids
Geometry
Generalization

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

@article{2c1fe92fb72749dfbf026acf779d4967,
title = "Ergodicity and Hopf–Lax–Oleinik formula for fluid flows evolving around a black hole under a random forcing",
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.",
keywords = "Burgers equation, Ergodicity, Hopf–Lax–Oleinik formula, One-force-one-solution principle, Random forcing, Schwarzschild black hole",
author = "Yuri Bakhtin and LeFloch, {Philippe G.}",
year = "2018",
month = "12",
day = "1",
doi = "10.1007/s40072-018-0119-8",
language = "English (US)",
volume = "6",
pages = "746--785",
journal = "Stochastics and Partial Differential Equations: Analysis and Computations",
issn = "2194-0401",
publisher = "Springer New York",
number = "4",

}

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 -