Absence of local anomalous dissipation and local energy balance in 2D incompressible flows away from the boundary

For the 2D Navier-Stokes equations with no-slip boundary condition, we consider the issue of whether anomalous dissipation away from the boundary vanishes. In particular, we show that such vanishing occurs if $u^{\nu}$ is uniformily bounded in the Onsager supercritical space $L^{1+}_{t}L^{\infty}_{x,loc}$ with appropriate bounds on the initial conditions. Our method involves arguments from \cite{AD23} and \cite{CW23} involving localization via modulation, together with vorticity energy type estimates inspired by \cite{CFLS16} and estimates involving $L^2$-based structure functions inspired by \cite{DP25dissconc}. Next we show that the aforementioned setting produces convergence to an Euler solution with its large scale approximation satisfying a local energy balance equation. Notably, we do not assume any uniform-in-viscosity bounds on the pressure. The large scale approximation has been introduced in \cite{PGLR18} in the context of partial regularity of the 3D Navier-Stokes equations, yet to the best of our knowledge this is the first time it has been considered in the context of inviscid limits.