Random attractors of stochastic partial differential equations: A smooth approximation approach

ABSTRACT

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.

KEYWORDS:

• Stochastic partial differential equations
• random dynamical systems
• random attractor
• upper semi-continuity
• Wong-Zakai approximation

MATHEMATICS SUBJECT CLASSIFICATION:

• 37L55
• 35R60
• 58B99
• 35L20

Funding

Xianming Liu acknowledges the support from NSFC 11301197, 11271013, and Specialized Research Fund for the Doctoral Program of Higher Education (20110142120036). Meihua Yang acknowledges the support of NSFC grant (11571125).