Approximation of invariant measures of dissipative dynamical systems on thin domains

An abstract method is presented to show that upper semicontinuity of invariant measures of dissipative dynamical systems on thin domains. The abstract method presented can be used to many physical systems. As an example, we consider reaction-diffusion equations on thin domains. To this end, we first show the existence of invariant measures of the equations in a bounded domain in \(\mathbb {R}^{n+1}\) ${\mathbb{R}}^{n+1}$ which can be viewed as a perturbation of a bounded domain in \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . We then prove that any limit of invariant measures of the perturbed systems must be an invariant measure of the limiting system when the thin domains collapses.