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We study a dynamical large deviation principle for global solutions to thethree-dimensional Landau-Lifschitz-Navier-Stokes equations with spatiallycorrelated noise, in a scaling regime where the noise intensity and correlationlength go to zero simultaneously. Paralleling the classical Leray theory, thesolutions are defined globally in time and satisfy a path-by-path energyinequality. We relate classes of solutions on which the large deviations upperand lower bounds match to the deterministic energy equality, generalising theresult of Lions-Ladyzhenskaya, and relate potential failure of the energyequality to violations of the large deviations lower bound without restriction.Finally, we consider the large deviations of the local-in-time strong solution,and show that the solution can rapidly lose regularity with negligiblelarge-deviations cost.

Landau-Lifshitz-Navier-Stokes Equations: Large Deviations and Relationship to The Energy Equality