Exact conservation and the Onsager threshold: a discrete exterior calculus theory for incompressible Navier-Stokes Equations

TL;DR

This paper introduces a structure-preserving discretisation of incompressible Euler and Navier-Stokes equations using discrete exterior calculus, ensuring exact conservation and addressing key regimes including smooth, weak, measure-valued, and dissipative solutions.

Contribution

It develops a rigorous, structure-preserving discretisation framework that enforces exact algebraic conservation, influencing solution classes and convergence properties across various fluid dynamics regimes.

Findings

Convergence rates depend on mesh regularity and reconstruction accuracy.

Limits in the inviscid measure-valued regime are conservative and align with Onsager's threshold.

Energy conservation at the discrete level excludes dissipative solutions, highlighting a key structural property.

Abstract

We develop a rigorous theory for a structure-preserving discretisation of the incompressible Euler and Navier--Stokes equations, based on discrete exterior calculus on prismatic Delaunay--Voronoi meshes over closed Riemannian manifolds. The central result is a selection principle: exact algebraic conservation at the discrete level is not merely a fidelity property but rules out entire classes of weak solutions that other discretisations reach unconditionally. We establish this in four regimes. \emph{Smooth solutions}: convergence at rate $\mathcal{O}(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|^{\beta_d})$, uniformly in viscosity $\nu \ge 0$, with $\beta_3 = 0$ and $\beta_2 = 1$; first order on general meshes and second order on meshes with centroid proximity and reconstruction symmetry. \emph{Leray--Hopf weak regime}: subsequential $L^2$ limits are weak solutions of the viscous system. \emph{Inviscid measure-valued regime}: limits are conservative measure-valued Euler solutions; their concentration defect vanishes above the Onsager threshold $\alpha > 1/3$ \emph{provided the discrete solutions admit a uniform $C^{0,\alpha}$ bound there}. \emph{Dissipative regime}: no subsequence converges to an energy-dissipating Euler solution at any regularity, a structural exclusion that follows from exact discrete energy conservation and distinguishes the scheme. The gap $1/3 < \alpha < 1$, where energy conservation and defect-free convergence hold but uniqueness remains open, isolates the central open problem of inviscid fluid dynamics.