Onsager's conjecture and anomalous dissipation on domains with boundary

TL;DR

This paper establishes localized regularity conditions for energy conservation in Euler solutions with boundary, and shows vanishing viscous dissipation in the inviscid limit for Navier-Stokes solutions under boundary layer assumptions.

Contribution

It introduces new boundary regularity criteria ensuring energy conservation and vanishing dissipation in the inviscid limit, extending Onsager's conjecture to bounded domains.

Findings

Energy conservation holds under Besov regularity and boundary conditions.

Viscous dissipation vanishes in the inviscid limit with boundary layer assumptions.

Strong convergence to Euler solutions is achieved under the proposed conditions.

Abstract

We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain $\Omega\subset \mathbb{R}^d$, $d\ge 2$, with boundary. In the bulk of fluid, we assume Besov regularity of the velocity $u\in L^3(0,T;B_{3}^{1/3, c_0})$. On an arbitrary thin neighborhood of the boundary, we assume boundedness of velocity and pressure and, at the boundary, we assume continuity of wall-normal velocity. We also prove two theorems which establish that the global viscous dissipation vanishes in the inviscid limit for Leray--Hopf solutions $u^\nu$ of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a thin boundary layer of width $O(\nu^{\min\{1,\frac{1}{2(1-\sigma)}\}})$ when $u\in L^3(0, T; B_3^{\sigma, c_0})$ in the interior for any $\sigma\in [1/3,1]$. The first theorem assumes continuity of the velocity in the boundary layer whereas the second assumes a condition on the vanishing of energy dissipation within the layer. In both cases, strong $L^3_tL^3_{x,loc}$ convergence holds to a weak solution of the Euler equations. Finally, if a strong Euler solution exists in the background, we show that equicontinuity at the boundary within a $O(\nu)$ strip alone suffices to conclude the absence of anomalous dissipation.