The Navier-Stokes equations and weak Herz spaces

Abstract

In this paper, we discuss the Cauchy problem for Navier-Stokes equations in homogeneous weak Herz spaces $W{\dot{K}}^\alpha_{p,q}({\mathbb{R^{\textit{n}}}})$. More precisely, we construct the solution in the class $L^\infty(0,T; W{\dot{K}}^\alpha_{p,q})$ with the initial data in $W{\dot{K}}^\alpha_{p,q}$. Further, we consider the blow-up phenomena of time-local solutions and the uniqueness of global solutions with large initial data in $W{\dot{K}}^\alpha_{p,q}$. Also, we give several embeddings of weak Herz spaces into homogeneous Besov spaces $B^{-\alpha}_{p,\infty}({\mathbb{R^{\textit{n}}}})\ (\alpha >0),$ or $bmo^{-1}({\mathbb{R^{\textit{n}}}})$.