An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

Abstract

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 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 dynamic boundary conditions. The purpose of this article 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.

Keywords:

• Microscopic mechanism on boundary
• Nonlinear system under uncertainty
• Random attractor
• Random dynamical system
• Stochastic dynamic boundary condition
• Stochastic partial differential equations (SPDES)

Mathematics Subject Classification:

• Primary 60H15, 37L55
• Secondary 35R60, 37H10

We thank Alain Miranville and Dirk Blomker for helpful comments. This work was partly supported by the NSF Grants DMS-0209326 & DMS-0542450.