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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2506.01573 |
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| _version_ | 1866908389606424576 |
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| author | Kobayashi, Takayuki Nakasato, Ryosuke |
| author_facet | Kobayashi, Takayuki Nakasato, Ryosuke |
| contents | 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, $P'(ρ_*)=0$, 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 $L^p$-$L^1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces Kobayashi, Takayuki Nakasato, Ryosuke Analysis of PDEs 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, $P'(ρ_*)=0$, 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 $L^p$-$L^1$ 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. |
| title | Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.01573 |