随机动力边界条件对Cahn-Hilliard系统演化的影响
An impact of stochastic dynamic boundary conditions on the evolution of the Cahn-Hilliard system
摘要：
非线性系统经常受到随机因素的影响。有时噪声通过物理边界进入系统，这导致随机动态边界条件。与静态边界条件相反，动态边界条件涉及边界上系统状态变量的时间导数和空间导数。虽然随机\emph{static}（Neumann或Dirichet型）边界条件已被应用于随机偏微分方程，但对随机\emph{dynamic}边界条件的动力学影响知之甚少。本文研究了材料科学中的Cahn-Hilliard方程在随机动态边界条件下的长期动力学行为。我们发现，随机吸引子的维数估计增加的系数的动态项的随机动态边界条件的减少。然而，随机吸引子的维数不受相应的随机静态边界条件的影响。
英文摘要：
Nonlinear systems are often subject to random influences. Sometimes the noise enters the system through physical boundaries and this leads to stochastic dynamic boundary conditions. A dynamic, as opposed to static, boundary condition involves the time derivative as well as spatial derivatives for the system state variables on the boundary. Although stochastic \emph{static} (Neumann or Dirichet type) boundary conditions have been applied for stochastic partial differential equations, not much is known about the dynamical impact of stochastic \emph{dynamic} boundary conditions. The purpose of this paper is to study possible impacts of stochastic dynamic boundary conditions on the long term dynamics of the Cahn-Hilliard equation arising in the materials science. We show that the dimension estimation of the random attractor increases as the coefficient for the dynamic term in the stochastic dynamic boundary condition decreases. However, the dimension of the random attractor is not affected by the corresponding stochastic static boundary condition.
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