  In this paper, we study the Cauchy problem to the 3D fractional compressible
isentropic generalized Navier-Stokes equations for viscous compressible fluid
with one Levy diffusion process. We obtain the existence and uniqueness of
global strong solution for small initial data by providing several commutators
via the Littlewood-Paley theory. Moreover, we derive $L^2$-decay rate for the
highest derivative of the strong solution without decay loss by using a
cancellation of a low-medium-frequency quantity. Our results improve the
results provided in [J. Differential Equations 377(2023): 369-417].
