Global regularity and sharp decay rates for the one-dimensional hypo-viscous compressible Navier-Stokes equations in critical Besov spaces

In this paper, we study the global regularity of the one-dimensional hypo-viscous compressible Navier–Stokes equations in the critical Besov spaces, where the initial data are close to the constant equilibrium state. We introduce an effective velocity, constructed as a combination of the density and the velocity, which is used to capture the high-frequency behavior of $a$. By performing elaborate high-frequency and low-frequency estimates together with a bootstrap argument, we obtain global existence for $\frac{1}{2}\le \beta <1$. In particular, when $\frac{1}{2}\le \beta <\frac{3}{4}$, we also prove the uniqueness of solutions. Furthermore, under an additional low-frequency assumption, we establish the optimal time decay rates.