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 , assuming that the solutions have norms for Besov space that are bounded in the L³-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form vanishing as if A consequence is that Onsager-type 'quasi-singularities' are required in the Leray solutions, even if the total energy dissipation vanishes in the limit , 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 the anomalous dissipation vanishes and the weak Euler solutions may be spatially 'rough' but conserve energy.