Besov space approach to the Navier-Stokes equations with the Neumann boundary condition in bounded domains

Abstract

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 $\Omega \subset {\mathbb R}^d$ with smooth boundary $\partial \Omega$ in terms of the Stokes operator $A=A_2$ with the Neumann boundary condition on $\partial\Omega$ in $L^2_{\sigma}(\Omega)$. Under some geometric assumption on $\Omega$, 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.