Space-time stationary solutions for the Burgers equation
Space-time stationary solutions for the Burgers equation
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DOI：
10.1090/s0894-0347-2013-00773-0
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发表时间：
2012-05
影响因子：
3.9
通讯作者：
Yuri Bakhtin;E. Cator;K. Khanin
中科院分区：
文献类型：
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作者：
Yuri Bakhtin;E. Cator;K. Khanin
在没有周期或任何其他紧性假设的情况下，我们构造了具有随机强迫的一维Burgers方程的时空定常解。更确切地说，对于时空中的齐次泊松点场给出的作用力，我们证明了在任何给定的平均速度下，存在唯一的整体解。这些整体解作为与柯西问题解相关的无限维动力系统的一点随机吸引子。整体解的概率分布定义了相应的马尔可夫过程的平稳分布。我们描述了一类广泛的初始柯西数据，对于它，马尔可夫过程的分布收敛于上述平稳分布。我们的全局解的构造是基于对行动极小化曲线领域的研究。我们证明了对于任意值的平均速度，概率为1，存在一个唯一的作用极小化曲线场，它始于所有的泊松点处。此外，对应于不同起始点的作用最小化曲线在有限时间内相互融合。
We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poissonian point field in space-time we prove that there is a unique global solution with any prescribed average ve- locity. These global solutions serve as one-point random attractors for the infinite-dimensional dynamical system associated to solutions to the Cauchy problem. The probability distribution of the global solutions de- fines a stationary distribution for the corresponding Markov process. We describe a broad class of initial Cauchy data for which the distribution of the Markov process converges to the above stationary distribution. Our construction of the global solutions is based on a study of the field of action-minimizing curves. We prove that for an arbitrary value of the average velocity, with probability 1 there exists a unique field of action-minimizing curves initiated at all of the Poissonian points. Moreover action-minimizing curves corresponding to different starting points merge with each other in finite time.