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 ω ⊂ ℝ^d, d ≥ 2, with boundary. In the bulk of uid, we assume Besov regularity of the velocity u ϵ L3(0, T;B1=3,c0 3 ). 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 uν of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a thin boundary layer of width O(ν min-1, 1/2(1-σ) }) when u ϵ L3(0, T;Bγ,c0 3 ) in the interior for any σ ϵ [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 L3t L3 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(v) strip alone suffices to conclude the absence of anomalous dissipation.