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

Abstract 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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 &amp; 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.