### 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 language | English (US) |
---|---|

Pages (from-to) | 193-238 |

Number of pages | 46 |

Journal | Journal of the American Mathematical Society |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the American Mathematical Society*,

*27*(1), 193-238. https://doi.org/10.1090/S0894-0347-2013-00773-0

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

Research output: Contribution to journal › Article

*Journal of the American Mathematical Society*, vol. 27, no. 1, pp. 193-238. https://doi.org/10.1090/S0894-0347-2013-00773-0

}

TY - JOUR

T1 - Space-time stationary solutions for the Burgers equation

AU - Bakhtin, Yuri

AU - Cator, Eric

AU - Khanin, Konstantin

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84885231705&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84885231705&partnerID=8YFLogxK

U2 - 10.1090/S0894-0347-2013-00773-0

DO - 10.1090/S0894-0347-2013-00773-0

M3 - Article

AN - SCOPUS:84885231705

VL - 27

SP - 193

EP - 238

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 0894-0347

IS - 1

ER -