Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $

In this paper, for the damped generalized incompressible Navier-Stokes equations on \begin{document}$ \mathbb{T}^{2} $\end{document} as the index \begin{document}$ \alpha $\end{document} of the general dissipative operator \begin{document}$ (-\Delta)^{\alpha} $\end{document} belongs to \begin{document}$ (0,\frac{1}{2}] $\end{document} , we prove the absence of anomalous dissipation of the long time averages of entropy. We also give a note to show that, by using the \begin{document}$ L^{\infty} $\end{document} bounds given in Caffarelli et al. [ 4 ], the absence of anomalous dissipation of the long time averages of energy for the forced SQG equations established in Constantin et al. [ 12 ] still holds under a slightly weaker conditions \begin{document}$ \theta_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{2}(\mathbb{R}^{2}) $\end{document} and \begin{document}$ f \in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2}) $\end{document} with some \begin{document}$ p>2 $\end{document} .