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⊂ Rᵈ, d≥ 2, with boundary. In the bulk of fluid, we assume Besov regularity of the velocity u∈ L³(0,T;B₃1/3, c₀). 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^ν 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∈ L³(0, T; B₃σ, c₀) in the interior for any σ∈ [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 L³ₜL³x,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(ν) strip alone suffices to conclude the absence of anomalous dissipation.