Global regularity and sharp decay rates to the 1D hypo-viscous compressible Navier-Stokes equations

In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium. Especially, we establish the global critical regularity in the Sobolev space $H^{\beta }$ with $\frac{1}{2}<\beta <1$. Furthermore, by bootstrap argument, Fourier splitting method and energy method, we then establish the optimal time decay rates under the extra low-frequency smallness assumption. We find the $L^2$ energy is self-closed, which motivates us to obtain the existence of global large solutions for initial data with high regularity. By a pure energy method, we also derive the optimal time decay rates when $\frac{1}{2}\le \beta <\frac{3}{4}$. We find a phenomenon that $\mathrm{\parallel }(a,u){\mathrm{\parallel }}_{L^2}$ still decays even if the initial data does not possess $L^2$ smallness. Notably, the low-frequency smallness assumption is removed in the case with $\frac{1}{2}\le \beta <\frac{3}{4}$.