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This paper is dedicated to the study of compressible Navier-Stokes equationsin $\mathbb{R}^d\ (d=2,3)$ with a fractional viscous term $(-\Delta)^{\alpha}$and $\alpha\in(\frac{1}{2},1)$. We establish the global existence of strongsolutions to this system with small initial data restricted in Sobolev spaces$H^{s+1}(\mathbb{R}^d)(s>\frac{d}{2})$ by providing several commutators andusing harmonic analysis tools (especially the Littlewood-Paley theory). Thisreduces the requirement of the regularity $H^{3\alpha+1}(\mathbb{R}^3)$ for theglobal existence in Wang-Zhang [41]. Additionally, we derive $L^2$-decay ratefor the highest derivative of the strong solution without decay loss by using acancellation of a low-medium-frequency quantity.

The global strong solution to compressible system with fractional viscous term