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The 2D Euler equations with random initial condition has been investigates by Albeverio and Cruzeiro (Commun Math Phys 129:431–444, 1990 ) and other authors. Here we prove existence of solutions for the associated continuity equation in Hilbert spaces, in a quite general class with LlogL densities with respect to the enstrophy measure.

Continuity equation in LlogL for the 2D Euler equations under the enstrophy measure