Onsager's Conjecture and Anomalous Dissipation on Domains with Boundary

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.