On the time-decay with the diffusion wave phenomenon of the solution to the compressible Navier-Stokes-Korteweg system in critical spaces

We consider the initial value problem of the compressible Navier-Stokes-Korteweg equations in the whole space $\mathbb{R}^d$ ($d \ge 2$). The purposes of this paper are to obtain the global-in-time solution around the constant equilibrium states $(\rho_*,0)$ and investigate the $L^p$-$L^1$ type time-decay estimates in a scaling critical framework, where $\rho_*>0$ is a constant. In addition, we study the diffusion wave property came from the wave equation with strong damping for the solution with the initial data belonging to the critical Besov space. The key idea of the proof is the derivation of the time-decay for the Fourier-Besov norm with higher derivatives by using $L^1$-maximal regularity for the perturbed equations around $(\rho_*,0)$.