INVISCID LIMIT FOR THE DAMPED GENERALIZED INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON T2

In this paper, for the damped generalized incompressible Navier-Stokes equations on T-2 as the index alpha of the general dissipative operator (-Delta)(alpha) belongs to (0,1/2], 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 L-infinity 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 theta(0) is an element of L-1(R-2 ) boolean AND L-2 (R-2) and f is an element of L-1(R-2)boolean AND L-p(R-2) with some p > 2.