An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes

Abstract

We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus T d , assuming that the solutions have norms for Besov space B σ,∞ 3 (T d ), σ ∈ (0, 1], that are bounded in the L 3 -sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form O(ν (3σ−1)/(σ+1) ), vanishing as ν → 0 if σ > 1/3. A consequence is that Onsagertype "quasi-singularities" are required in the Leray solutions, even if the total energy dissipation vanishes in the limit ν → 0, as long as it does so sufficiently slowly. We also give two sufficient conditions which guarantee the existence of limiting weak Euler solutions u which satisfy a local energy balance with possible anomalous dissipation due to inertial-range energy cascade in the Leray solutions. For σ ∈ (1/3, 1) the anomalous dissipation vanishes and the weak Euler solutions may be spatially "rough" but conserve energy.

Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit

“…Under the assumption of uniformly-in-viscosity positive local scaling exponents for the second-order structure functions, we have established the weak inviscid limit on arbitrary domains with smooth boundary. The imposed condition is no stronger than what is required on domains without boundaries [20,21,22] and is consistent with available data of turbulent flow on domains both with and without solid walls. As computing power improves and more experiments are run, the validity of (12) can continue to be checked over longer scale ranges and higher Reynolds numbers.…”

“…We say that weak inviscid limit holds if u ν converges (along a subsequence) weakly in L 2 t L 2 x to a weak solution u of the Euler equations as ν → 0. Without boundaries, [20,21,22] show that any suitable fractional degree of regularity implies the weak inviscid limit. On domains with boundary, the situation is more subtle and less is understood.…”

Convex integration and phenomenologies in turbulence

“…7 7 It is worth emphasizing that ε > 0 implies that the sequence of Navier-Stokes solutions {v ν }ν>0, say of Leray-Hopf kind, cannot remain uniformly bounded (with respect to ν) in the space L 3 t B s 3,∞,x for any s > 1 /3. In fact, in [59] it is shown that even if ε = 0, but the rate of vanishing of ε ν is slow, say lim infν→0 log(ε ν ) log(ν) = α ∈ (0, 1], then the sequence of Leray solutions v ν cannot remain uniformly in the space L 3 t B s 3,∞,x with s > 1+α 3−α . Thus, experimental evidence robustly points towards Euler singularities.…”

Energy Equality in Compressible Fluids with Physical Boundaries

“…An alternative proof of Shinbrot's result can be found in [39]. New types of conditions have been obtained recently, including Besov-type regularity conditions [6,13], weak-in-time with optimal Onsager spatial regularity conditions [7], new L p t L q x conditions in combination with low dimensionality of the singular set [26], to name a few. For inhomogeneous incompressible flows Leslie-Shvydokoy [25] proved the energy equality in Besov spaces.…”