Well-posedness And Large Deviation Principle For SPDEs With Locally Monotone Coefficients

In this paper,the well-posedness for a class of SPDEs is proved in case of the mul-tiplicative noise.Using the weak convergence approach,we also prove the large devia-tion principle for a class of SPDEs with locally monotone coefficients under variational framework(the Freidlin-Wentzell type large deviation principle).For the multiplicative noise case,the well-posedness of the SPDEs can be obtained by proving the existence of martingale solutions and pathwise uniqueness.Based on this,the large deviation principle can be obtained in case of the additive noise and the multiplicative noise.Because of the equivalent between the large deviation principle and the Laplace principle,assuming some regularity w.r.t.the time variable on the diffusion coefficient(for the multiplicative noise case),the large deviation principle is proved by using the stochastic control and weak con-vergence approach.Therefore,some former results obtained in[10,11,32,46,51,53,62]are generalized and improved in this paper.The content of the thesis consists of six parts as follows;In Chapter 1,we introduce the research background and recent development of SPDEs and LDP,and the main results of our paper is also briefly introduced.In Chapter 2,we mainly give some knowledge for SPDEs and the large deviation principle.In Chapter 3,the large deviation principle is proved in case of the additive noise case.In Chapter 4,the uniqueness and existence of strong solutions for SPDEs is proved in case of the multiplicative noise.In Chapter 5,the large deviation principle for the multiplicative noise case is proved.In Chapter 6,the main result in Chapter 5 is applied to some concrete models to verify the large deviation principle.