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 T^d, assuming that the solutions have norms for Besov space B^σ,&inf; ₃ (T^d), σ &insin; (0, 1], that are bounded in the L³-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form O(v(3σ-1)/(σ+1)), vanishing as v ← 0 if σ > 1/3. A consequence is that Onsager-type 'quasi-singularities' are required in the Leray solutions, even if the total energy dissipation vanishes in the limit v ← 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 &insin; (1/3, 1) the anomalous dissipation vanishes and the weak Euler solutions may be spatially 'rough' but conserve energy.