An Onsager Singularity Theorem for Leray Solutions of Incompressible Navier-Stokes

We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus ${\mathbb{T}}^d$, assuming that the solutions have norms for Besov space $B_3^{\sigma ,\mathrm{\infty }}({\mathbb{T}}^d),$ $\sigma \in (0,1],$ that are bounded in the $L^3$-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form $O({\nu }^{(3\sigma -1)\mathrm{/}(\sigma +1)}),$ vanishing as $\nu \to 0$ if $\sigma >1\mathrm{/}3.$ A consequence is that Onsager-type “quasi-singularities” are required in the Leray solutions, even if the total energy dissipation vanishes in the limit $\nu \to 0$, 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 $\sigma \in (1\mathrm{/}3,1)$ the anomalous dissipation vanishes and the weak Euler solutions may be spatially “rough” but conserve energy.