Large Deviation Principle For Stochastic Partial Differential Equations With Locally Monotone Coefficients

In this paper,using the weak convergence approach,we establish the large devia-tion principle for a class of SPDEs with locally monotone coefficients.The main result obtained in this thesis not only covers some known results obtained in[12,54,35],but also can be applied to all SPDE models contained in[37],including some new ex-amples such as stochastic generalized Burgers equations,2D stochastic Navier-Stokes equations and stochastic power law equations.In this paper,we mainly use the weak convergence approach to prove the large deviation principle.Although the main advantage of weak convergence approach is to avoid some technical exponential probability estimates in classical discretization method,there are still many technical difficulties during the implement of weak conver-gence approach.The main reason is the coefficients of SPDEs are nonlinear operators,which are well-defined only under a Gelfand triple.Hence we have to handle many estimates involving different norms,some approximation techniques(e.g.finite dimen-sional approximation and truncation techniques)are used in the proof.The content of this thesis consists of following four chapters:In Chapter 1,we mainly introduce the development of stochastic partial differen-tial equations and the investigation of large deviation principle,and the main result of our work is also briefly introduced.In Chapter 2,we mainly present some preliminary knowledge related to our work.In Chapter 3,we prove the large deviation principle for a class of SPDEs with locally monotone coefficients.In Chapter 4,we apply our main result to some concrete examples to establish the large deviation principle for those models.