Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces

Abstract

We consider the initial-value problem in the $d$-dimensional Euclidean space ${\mathbb{R}}^d$ $(d\ge 3)$ for the compressible Navier-Stokes-Korteweg equations under the zero sound speed case (namely, $P^{\prime }({\rho }_{\ast })=0$, where $P=P(\rho )$ stands for the pressure). The system is well-known as the Diffuse Interface model describing the motion of a vaper-liquid mixture in a compressible viscous fluid. The purposes of this paper are to obtain the global-in-time solution around the constant equilibrium states $({\rho }_{\ast },0)$ $({\rho }_{\ast }>0)$ satisfying the estimate on the analyticity as established by Foias-Temam (1989), and investigate the $L^p$-$L^1$ type time-decay estimates in scaling critical settings based on Fourier-Herz spaces. In addition, we also derive the first order asymptotic formula with higher derivatives for solutions as the application of the analyticity.