On the Endpoint Regularity in Onsager's Conjecture

Abstract

Onsager's conjecture states that the conservation of energy may fail for $3D$ incompressible Euler flows with Hölder regularity below $1/3$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the $3D$ incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,1/3)$. Our construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of \cite{IOnonpd} to modify the convex integration scheme. We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov-Obukhov scaling for any $p>3$ if their singular supports have space-time Lebesgue measure zero.

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References in corpus (6)

• Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations
• 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
• An Intermittent Onsager Theorem
• Convex integration for the Monge-Ampère equation in two dimensions

Cited by in corpus (10)

• Sharp nonuniqueness for the Navier-Stokes equations
• The energy equality for the Navier-Stokes equations in bounded domains
• Onsager's energy conservation for inhomogeneous Euler equations
• Finite energy weak solution of 2d Boussinesq equation with diffusive temperature
• Anomalous dissipation, anomalous work, and energy balance for smooth solutions of the Navier-Stokes equations
• Weak solutions of the three-dimensional hypoviscous elastodynamics with finite kinetic energy
• Non-conservative $H^{\frac{1}{2}-}$ weak solutions of the incompressible 3D Euler equations
• A Kato-type criterion for vanishing viscosity near the Onsager's critical regularity
• Onsager's Conjecture for the Incompressible Euler Equations in the Hölog Spaces $C_{\lambda }^{0,\alpha }(\bar{\Omega })$
• Hölder continuous weak solution of 2d Boussinesq equation with diffusive temperature