Space-time stationary solutions for the Burgers equation

Yuri Bakhtin, Eric Cator, Konstantin Khanin

Research output: Contribution to journalArticle

Abstract

We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poisson point field in space-time we prove that there is a unique global solution with any prescribed average velocity. These global solutions serve as one-point random attractors for the infinite-dimensional dynamical system associated with solutions to the Cauchy problem. The probability distribution of the global solutions defines a stationary distribution for the corresponding Markov process. We describe a broad class of initial Cauchy data for which the distribution of the Markov process converges to the above stationary distribution. Our construction of the global solutions is based on a study of the field of action-minimizing curves. We prove that for an arbitrary value of the average velocity, with probability 1 there exists a unique field of action-minimizing curves initiated at all of the Poisson points. Moreover, action-minimizing curves corresponding to different starting points merge with each other in finite time.

Original languageEnglish (US)
Pages (from-to)193-238
Number of pages46
JournalJournal of the American Mathematical Society
Volume27
Issue number1
DOIs
StatePublished - 2013

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Burgers Equation
Stationary Solutions
Global Solution
Markov processes
Space-time
Stationary Distribution
Markov Process
Probability distributions
Forcing
Curve
Siméon Denis Poisson
Dynamical systems
Infinite Dimensional Dynamical System
Random Attractor
Periodicity
Cauchy
Compactness
Cauchy Problem
Probability Distribution
Converge

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Space-time stationary solutions for the Burgers equation. / Bakhtin, Yuri; Cator, Eric; Khanin, Konstantin.

In: Journal of the American Mathematical Society, Vol. 27, No. 1, 2013, p. 193-238.

Research output: Contribution to journalArticle

Bakhtin, Yuri ; Cator, Eric ; Khanin, Konstantin. / Space-time stationary solutions for the Burgers equation. In: Journal of the American Mathematical Society. 2013 ; Vol. 27, No. 1. pp. 193-238.
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