arXiv subjects

Ryosuke Nakasato

Publications and source records attributed to Ryosuke Nakasato.

2 recordsLinked to original sources

arXiv2025

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

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, $P'(\rho_*)=0$, 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 $(\rho_*,0)$ $(\rho_*>0)$ 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.

math.AP↗

arXiv2025

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 $(\rho_*,0)$.

math.AP↗