### Abstract

A statistical theory is developed for the stochastic Burgers equation in the inviscid limit. Master equations for the probability density functions of velocity, velocity difference, and velocity gradient are derived. No closure assumptions are made. Instead, closure is achieved through a dimension reduction process; namely, the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field, here the shocks. Master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks, namely, the points of shock generation and collisions. The scaling laws of the structure functions are derived through the analysis of the master equations. Rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints. We also establish that the probability density function Q(ξ) of the velocity gradient decays as |ξ|^{-7/2} as ξ → -∞.

Original language | English (US) |
---|---|

Pages (from-to) | 852-901 |

Number of pages | 50 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 53 |

Issue number | 7 |

State | Published - 2000 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Statistical theory for the stochastic Burgers equation in the inviscid limit.** / E, Weinan; Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 53, no. 7, pp. 852-901.

}

TY - JOUR

T1 - Statistical theory for the stochastic Burgers equation in the inviscid limit

AU - E, Weinan

AU - Vanden Eijnden, Eric

PY - 2000

Y1 - 2000

N2 - A statistical theory is developed for the stochastic Burgers equation in the inviscid limit. Master equations for the probability density functions of velocity, velocity difference, and velocity gradient are derived. No closure assumptions are made. Instead, closure is achieved through a dimension reduction process; namely, the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field, here the shocks. Master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks, namely, the points of shock generation and collisions. The scaling laws of the structure functions are derived through the analysis of the master equations. Rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints. We also establish that the probability density function Q(ξ) of the velocity gradient decays as |ξ|-7/2 as ξ → -∞.

AB - A statistical theory is developed for the stochastic Burgers equation in the inviscid limit. Master equations for the probability density functions of velocity, velocity difference, and velocity gradient are derived. No closure assumptions are made. Instead, closure is achieved through a dimension reduction process; namely, the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field, here the shocks. Master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks, namely, the points of shock generation and collisions. The scaling laws of the structure functions are derived through the analysis of the master equations. Rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints. We also establish that the probability density function Q(ξ) of the velocity gradient decays as |ξ|-7/2 as ξ → -∞.

UR - http://www.scopus.com/inward/record.url?scp=0001532759&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001532759&partnerID=8YFLogxK

M3 - Article

VL - 53

SP - 852

EP - 901

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 7

ER -