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Statistical theory for the stochastic Burgers equation in the inviscid limit
Author(s) -
E W.,
Vanden Eijnden Eric
Publication year - 2000
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(200007)53:7<852::aid-cpa3>3.0.co;2-5
Subject(s) - mathematics , inviscid flow , burgers' equation , limit (mathematics) , closure (psychology) , shock (circulatory) , probability density function , mathematical analysis , master equation , realizability , statistical physics , statistical theory , classical mechanics , physics , statistics , partial differential equation , law , quantum mechanics , medicine , algorithm , political science , quantum
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 → − ∞. © 2000 John Wiley & Sons, Inc.