Besov space approach to the Navier-Stokes equations with the Neumann boundary condition in bounded domains
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arXiv
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2026
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| _version_ | 1866918374164922368 |
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| author | Iwabuchi, Tsukasa Kozono, Hideo |
| author_facet | Iwabuchi, Tsukasa Kozono, Hideo |
| contents | Based on the analysis by Iwabuchi-Matsuyama-Taniguchi (2019), we first introduce our framework of Besov spaces $\dot B^s_{p, q}$ on the bounded domain $Ω\subset {\mathbb R}^d$ with smooth boundary $\partial Ω$ in terms of the Stokes operator $A=A_2$ with the Neumann boundary condition on $\partialΩ$ in $L^2_σ(Ω)$. Under some geometric assumption on $Ω$, we establish $L^p-L^q$ type estimates of the semi-group $\{e^{-tA}\}_{t \ge 0}$ in $\dot B^s_{p, q}$ and prove a local well-posedness of the Navier-Stokes equations with the initial data in $\dot B^{-1+\frac dp }_{p, q}$ for $d < p < \infty$ and $1 \le q \le \infty$. Since $d < p$, we have $L^{d, \infty} \subset \dot B^{-1+\frac dp }_{p, \infty}$ so that our space for well-posedness is larger than any other previous one in bounded domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05779 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Besov space approach to the Navier-Stokes equations with the Neumann boundary condition in bounded domains Iwabuchi, Tsukasa Kozono, Hideo Analysis of PDEs 76D05, 35Q30 Based on the analysis by Iwabuchi-Matsuyama-Taniguchi (2019), we first introduce our framework of Besov spaces $\dot B^s_{p, q}$ on the bounded domain $Ω\subset {\mathbb R}^d$ with smooth boundary $\partial Ω$ in terms of the Stokes operator $A=A_2$ with the Neumann boundary condition on $\partialΩ$ in $L^2_σ(Ω)$. Under some geometric assumption on $Ω$, we establish $L^p-L^q$ type estimates of the semi-group $\{e^{-tA}\}_{t \ge 0}$ in $\dot B^s_{p, q}$ and prove a local well-posedness of the Navier-Stokes equations with the initial data in $\dot B^{-1+\frac dp }_{p, q}$ for $d < p < \infty$ and $1 \le q \le \infty$. Since $d < p$, we have $L^{d, \infty} \subset \dot B^{-1+\frac dp }_{p, \infty}$ so that our space for well-posedness is larger than any other previous one in bounded domains. |
| title | Besov space approach to the Navier-Stokes equations with the Neumann boundary condition in bounded domains |
| topic | Analysis of PDEs 76D05, 35Q30 |
| url | https://arxiv.org/abs/2603.05779 |