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arXiv · 2506.01573

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

Takayuki Kobayashi·Ryosuke Nakasato
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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, $P'(\rho_*)=0$, 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.

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Takayuki Kobayashi, Ryosuke Nakasato. 2025-06-02. Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces. https://doi.org/10.1016/j.nonrwa.2026.104734

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