随机非线性反应扩散系统的自相似性和吸引性
Self similarity and attraction in stochastic nonlinear reaction-diffusion systems
摘要：
相似解在许多科学领域中起着重要的作用：我们在这里考虑随机动力学中的相似性。重要的问题不仅是随机相似性的存在，而且是否相似的解决方案是动态吸引的，如果是，什么特解系统的发展。通过重铸一类随机偏微分方程的形式，其中随机中心流形理论可以应用，我们解决这些问题，在这个类。为了确定性，研究了Burgers方程在随机力作用下的自相似性。在适当的假设下，构造了随机Burgers方程的平稳解，得到了该方程的随机自相似解的存在性.并证明了该算法的自相似解的渐近收敛。其次，在更一般的随机反应扩散系统中，随机中心流形理论提供了一个框架来构造相似解，确认其相关性，并确定任何紧致初始条件的正确解。第三，我们认为，动态移动的空间原点和动态拉伸的时间，提高了描述的随机相似性。最后，一个非常简单的湍流混合模型的应用程序显示如何异常波动可能会出现在涡扩散。我们讨论的技术和结果应适用于广泛的随机相似性问题。
英文摘要：
Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of stochastic PDEs in a form to which stochastic centre manifold theory may be applied we resolve these issues in this class. For definiteness, a first example of self-similarity of the Burgers' equation driven by some stochastic forced is studied. Under suitable assumptions, a stationary solution is constructed which yields the existence of a stochastic self-similar solution for the stochastic Burgers' equation. Furthermore, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems stochastic centre manifold theory provides a framework to construct the similarity solution, confirm its relevance, and determines the correct solution for any compact initial condition. Third, we argue that dynamically moving the spatial origin and dynamically stretching time improves the description of the stochastic similarity. Lastly, an application to an extremely simple model of turbulent mixing shows how anomalous fluctuations may arise in eddy diffusivities. The techniques and results we discuss should be applicable to a wide range of stochastic similarity problems.
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