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 ${\dot{B}}_{p,q}^s$ 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 $L_{\sigma }^2(\Omega )$. Under some geometric assumption on $\Omega$, we establish $L^p-L^q$ type estimates of the semi-group $\{e^{-tA}{\}}_{t\ge 0}$ in ${\dot{B}}_{p,q}^s$ and prove a local well-posedness of the Navier-Stokes equations with the initial data in ${\dot{B}}_{p,q}^{-1+\frac{d}{p}}$ for $d<p<\infty$ and $1\le q\le \infty$. Since $d<p$, we have $L^{d,\infty }\subset {\dot{B}}_{p,\infty }^{-1+\frac{d}{p}}$ so that our space for well-posedness is larger than any other previous one in bounded domains.