Freidlin-Wentzell Type Large Deviation Principle for Multiscale Locally Monotone SPDEs

This work is concerned with the Freidlin-Wentzell type large deviation principle for a family of multi-scale quasi-linear and semi-linear stochastic partial differential equations (SPDEs) with small multiplicative noise under the generalized variational setting, which extend several existing works to the multi-scale process. Employing the weak convergence method developed by Dupuis and Ellis \cite{DE} and Khasminskii's time discretization approach \cite{K}, the Laplace principle for SPDEs will be derived, which is equivalent to the large deviation principle. In particular, in this paper, we do not assume any compactness of the embedding on the Gelfand triple we considered in order to deal with the case of bounded and unbounded domains in some concrete models. Our main results are applicable to a wide family of SPDEs such as stochastic porous media equations, stochastic fast-diffusion equations, stochastic 2D hydrodynamical type models, stochastic p-Laplace equations, stochastic power law fluid equations and stochastic Ladyzhenskaya models.