TY - JOUR

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

AU - E, Weinan

AU - Vanden Eijnden, Eric

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2000/7

Y1 - 2000/7

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 → -∞.

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U2 - 10.1002/(SICI)1097-0312(200007)53:7<852::AID-CPA3>3.0.CO;2-5

DO - 10.1002/(SICI)1097-0312(200007)53:7<852::AID-CPA3>3.0.CO;2-5

M3 - Article

AN - SCOPUS:0001532759

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 -