We show that stationary statistical properties for uniformlydissipative dynamical systems are upper semi-continuous underregular perturbation and a special type of singular perturbationin time of relaxation type. The results presented are applicableto many physical systems such as the singular limit of infinitePrandtl-Darcy number in the Darcy-Boussinesq system forconvection in porous media, or the large Prandtl asymptotics forthe Boussinesq system.
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Xiaoming Wang (2008) studied this question.