Statistical theory for the stochastic Burgers equation in the inviscid limit

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)852-901
Number of pages50
JournalCommunications on Pure and Applied Mathematics
Volume53
Issue number7
StatePublished - 2000

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Inviscid Limit
Burgers Equation
Stochastic Equations
Shock
Master Equation
Gradient
Probability density function
Closure
Decay
Tail Probability
Realizability
Dimension Reduction
Structure-function
Scaling Laws
Scaling laws
Velocity Field
Collision
Statistics
Term

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.

In: Communications on Pure and Applied Mathematics, Vol. 53, No. 7, 2000, p. 852-901.

Research output: Contribution to journalArticle

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