Onsager critical solutions of the forced Navier-Stokes equations

Abstract

We answer positively to [3]*Question 2.4 by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space, for any fixed ε > 0. Our construction combines ideas of [3] and [5].1. Introduction. The forced Navier-Stokes equations on the 3-dimensional torus T 3 ≃ R 3 /Z 3 are given byforce that may depend on ν. When ν = 0 the Navier-Stokes equations (NS) reduce to the forced Euler equations

Scaling laws and exact results in turbulence *

“…In other words, the dissipative length scale ℓ D used in [23] is not the one predicted by Kolmogorov's theory. Therefore, we revisit this question and prove in corollary 1.6 that in fact (1.3) holds for ℓ D = ν 3 4 − for the solutions constructed in [3]. More generally, we show that the choice of dissipative length scale ℓ D can be made using knowledge of the second order structure function exponent, or in this setting, the number ζ 2 such that the sequence u ν enjoys uniform bounds in…”

“…x , for which 1 The smooth forcings f ν depend on ν and lose some regularity in the limit ν → 0, although they remain bounded in L 1+ t C 0+ x ; this bound rules out the possibility of anomalous dissipation for the heat equation [3].…”

“…The second motivation for this work lies in recent results of Bruèet al [3] and Hofmanovaet al [23]. The former constructs a sequence of Leray weak solutions {u ν } to the forced 1 3D Navier-Stokes equations, uniformly bounded in…”

“…regularity of the example from [3] in order to obtain the correct dissipative length scale. We can similarly treat the versions of (1.3) for the 4 15 and 4 5 laws (the former of which is not mentioned in [1,23], although it may be treated using the same ideas).…”