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

2025/06/02 by Kobayashi, Takayuki, Nakasato, Ryosuke
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.01573

Abstract

We consider the initial-value problem in the d-dimensional Euclidean space ℝ^d (d ≥ 3) for the compressible Navier-Stokes-Korteweg equations under the zero sound speed case (namely, P'(ρ_*)=0, where P=P(ρ) 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 (ρ_*,0) (ρ_*>0) satisfying the estimate on the analyticity as established by Foias-Temam (1989), and investigate the L^p-L¹ 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.

Related

• Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition
• On the time-decay with the diffusion wave phenomenon of the solution to the compressible Navier-Stokes-Korteweg system in critical spaces
• Global existence and analyticity of L^p solutions to the compressible fluid model of Korteweg type
• Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity
• On the \mathcal R-boundedness of solution operators for a compressible fluid model of Korteweg type in general domains