Exact conservation and the Onsager threshold: a discrete exterior calculus theory for incompressible Navier-Stokes Equations
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2026
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| author | Korn, Peter |
| author_facet | Korn, Peter |
| contents | 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 $\mathcal{O}(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|^{β_d})$, 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. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13048 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact conservation and the Onsager threshold: a discrete exterior calculus theory for incompressible Navier-Stokes Equations Korn, Peter Analysis of PDEs Numerical Analysis Primary~65M08, 65M12, 35Q30, Secondary~35Q31, 35D30, 58A12, 76D05, 76M12, 76M30 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 $\mathcal{O}(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|^{β_d})$, 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. 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. |
| title | Exact conservation and the Onsager threshold: a discrete exterior calculus theory for incompressible Navier-Stokes Equations |
| topic | Analysis of PDEs Numerical Analysis Primary~65M08, 65M12, 35Q30, Secondary~35Q31, 35D30, 58A12, 76D05, 76M12, 76M30 |
| url | https://arxiv.org/abs/2605.13048 |