Invariant measures for Burgers equation with stochastic forcing
Invariant measures for Burgers equation with stochastic forcing
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DOI：
10.2307/121126
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发表时间：
2000-05
影响因子：
4.9
通讯作者：
E. Weinan;K. Khanin;A. Mazel;Y. Sinai
中科院分区：
文献类型：
--
作者：
E. Weinan;K. Khanin;A. Mazel;Y. Sinai
本文研究了以下Burgers方程du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t)，其中f(x,t)=dF/dx（x,t）是一个随机强迫函数，它在x中是周期性的，在t中是白噪声的。我们通过建立“一力一解”原理证明了不变测度的存在性和唯一性，即对于几乎每一个力的实现，在时间区间（-infty），+ 50)，这个解以同样的力吸引所有其他解。这是通过研究所谓的片面最小化来实现的。我们还详细描述了平稳解的结构和正则性。特别地，我们证明了在强迫的一些非简并条件下，对于平稳解几乎肯定存在唯一的主震和唯一的全局最小值。此外，全局最小化器是底层特征系统的双曲轨迹。
In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some non-degeneracy conditions on the forcing, that almost surely there is a unique main shock and a unique global minimizer for the stationary solutions. Furthermore the global minimizer is a hyperbolic trajectory of the underlying system of characteristics.