Large Deviations of Landau-Lifschitz-Navier-Stokes & the Energy Equality

The Landau-Lifschitz-Navier-Stokes equations are stochastic partial differential equations, which introduce a stochastic forcing term to describe the macroscopic fluctuations away from the deterministic Navier-Stokes equations. We consider the large deviations of a suitable solution theory in a scaling regime where the noise intensity and correlation length go to zero simultaneously, with a coupled rate. We show that the large deviations reproduce those of the lattice gas studied by Quastel and Yau on sufficiently integrable fluctuations. The large deviation rate function is connected to weak-strong uniqueness and the validity of the energy equality for forced Navier-Stokes with forcing in the Leray class $L^2_tH^{-1}_x$, and there is no possibility of extending the theory into stronger regularity classes or integrability conditions.