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

14 pgs; v2: reorganized main results and added additional technical details to proofs, v3 accepted in Nonlinearity

References in corpus (6)

• Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains
• Onsager's Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit
• Onsager's conjecture and anomalous dissipation on domains with boundary
• Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit
• Remarks on high Reynolds numbers hydrodynamics and the inviscid limit
• Turbulent Cascade Direction and Lagrangian Time-Asymmetry

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