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

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 $\OO(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|^{β_d})$, uniformly in viscosity $ν\ge 0$, with $β_3 = 0$ and $β_2 = 1$; 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 $α> 1/3$ \emph{provided the discrete solutions admit a uniform $C^{0,α}$ 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. % from all Galerkin and finite-volume methods. The gap $1/3 < α< 1$, where energy conservation and defect-free convergence hold but uniqueness remains open, isolates the central open problem of inviscid fluid dynamics.