Well-posedness And Ergodicity For A Class Of SPDEs

The study of Stochastic partial differential equations involves the theory of probability,partial differential equation,infinite dimensional analysis and so on.At the same time it plays an important role in many applications such as physics,chemistry,biology and economics.There has been a wide spread interest and attention in the research of SPDEs,which have been one of the most active research areas in the probability theory.In this paper,the first main result is to prove the well-posedness of stochastic MHD equations by using generalized variational framework of SPDEs[50,51].The second main result is to investigate the ergodicity,contractivity properties for the associated transition semigroups by using Harnack inequality[48,49,74,75] under the generalized growth condition with classical variational framework[41].Moreover,the exponential convergence of the transition semigroups and the existence of a spectral gap are also derived.As examples,except for stochastic generalized heat equations,stochastic porous media equations and stochastic p-Laplace equations,the main results can also be applied to those equations with higher order perturbation in the drift.The content of the thesis is divided into four parts as follows:In Chapter 1,we introduce the research background and recent development of SPDEs,the main results of our work are also briefly introduced.In Chapter 2,we introduce some preliminaries to our work.In Chapter 3,we prove the well-posedness of 2D stochastic MHD equations using the generalized variational framework.In Chapter 4,we prove contractivity properties for the associated transition semigroups for SPDEs with coefficients satisfying generalized growth condition.As applications,we illustrate main results by applying to concrete SPDE models.