Very weak solutions and the Fujita-Kato approach to the Navier-Stokes system in general unbounded domains

Abstract

We consider the instationary Navier-Stokes system in general unbounded domains \({\Omega \subset \mathbb{R}^{n}}\), \({n\geq 3}\), with smooth boundary and construct by the Fujita-Kato method mild solutions \({u\in L^{\infty}(0,T; \tilde{L}^{n}(\Omega))}\) with initial value \({u_0\in\tilde{L}^{n}(\Omega)}\). Here the classical \({L^n(\Omega)}\)–space is replaced by \({\tilde{L}^n(\Omega)}\) where for q > 2 the space \({\tilde{L}^q}\) is defined by \({L^q\cap L^2}\). Moreover, for suitable initial values we identify mild solutions in \({L^\infty(0,T;\tilde{L}^n(\Omega))}\) with very weak solutions in Serrin’s class \({L^r (0,T;\tilde{L}^q(\Omega))}\) where \({\frac{2}{r} + \frac{n}{q} =1}\), \({2< r < \infty}\).