Space-time statistical solutions of the incompressible Euler equations and Landau-Lifshitz fluctuating hydrodynamics

We study the infinite Reynolds limit of the solutions of the Landau-Lifshitz-Navier–Stokes equations for an incompressible fluid on a d -dimensional torus for d ⩾ 2 . These equations, which model thermal fluctuations in fluids, are given a standard physical interpretation as a low-wavenumber ‘effective field theory’, rather than as stochastic partial differential equations. We study solutions which enjoy some Besov regularity in space, as expected for initial data chosen from a driven, turbulent steady-state ensemble. The empirical basis for this regularity hypothesis, uniform in Reynolds number, is carefully discussed. Considering the initial-value problem for the Landau-Lifshitz equations, our main result is that the infinite Reynolds number limit is a space-time statistical solution of the incompressible Euler equations in the sense of Vishik & Fursikov. Such solutions are described by a probability measure on space-time velocity fields whose realizations are weak solutions of the incompressible Euler equations, each with the same prescribed initial data.