Inviscid Limit For Critical Quasi-geostrophic Equation In R~2

In this paper, we consider the inviscid limit of Generalized critical quasi-geostrophic equation in R2. First, we prove that absence of anomalous dissipation of energy for s-tationary equation. Then, we consider vanishing viscosity limit of long time averaged of solutions, we need the idea of C.Foias’s stationary statistical solution, we prove that, when the viscosity vanishes, stationary statistical solution of viscous forced 2D critical Q-G equations weak converge to stationary statistical solution of inviscid Q-G equation, and this limit solution obey the energy balance. Last, we use the energy balance prove that the rate of dissipation of energy long time averaged of solutions converges to zero...