Talagrand's transportation inequality for SPDEs with locally monotone drifts

The purpose of this paper is twofold. Firstly, we prove transportation inequalities ${\mathbf{T}}_{\mathbf{2}}(C)$ on the space of continuous paths with respect to the uniform metric for the law of the solution to a class of non-linear monotone stochastic partial differential equations (SPDEs) driven by the Wiener noise. Furthermore, we also establish the ${\mathbf{T}}_{\mathbf{1}}(C)$ property for such SPDEs but with merely locally monotone coefficients, including the stochastic Burgers type equation and stochastic $2$-D Navier-Stokes equation.