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

TL;DR

This paper investigates the energy dissipation behavior of Leray solutions to the incompressible Navier-Stokes equations in the inviscid limit, establishing conditions under which anomalous dissipation occurs and linking solution regularity to energy conservation.

Contribution

It provides an Onsager-type singularity theorem for Leray solutions, showing how solution regularity influences energy dissipation and the existence of weak Euler solutions with or without anomalous dissipation.

Findings

Energy dissipation bound: $O( u^{(3\sigma-1)/(\sigma+1)})$

Anomalous dissipation requires Onsager-type singularities for certain regularities

Weak Euler solutions can conserve energy if solution regularity exceeds a threshold

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 $B^{\sigma,\infty}_3({\mathbb T}^d),$ $\sigma\in (0,1],$ 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 $\nu\to0$ 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.