As one of the basic equations in fluid dynamics research, Navier-Stokes equation describes the dynamic balance of the force acting on any given region of liquid, reflecting the basic mechanical law of viscous fluid flow and has very important significance in fluid mechanics. As the generalization of the Navier-Stokes equation, the research on $g$-Navier-Stokes equation is flourishing in recent years. The dynamic properties of a class of autonomous $g$-Navier-Stokes systems with nonlinear dampness are studied. The global asymptotic compactness is proved by the prior estimation of solutions and the energy equation method on ${\bf R}^2$. The existence of global attractors about solvable semigroups is obtained, and the asymptotic smooth effect of solutions is analyzed. Some of the classical research results in recent years are further extended and improved, and the relevant research theories of $g$-Navier-Stokes equations are enriched.