Semigroup characterization of Besov type Morrey spaces and well-posedness of generalized Navier–Stokes equations

Abstract

The well-posedness of generalized Navier–Stokes equations with initial data in some critical homogeneous Besov spaces and in some critical Q spaces was known. In this paper, we establish a wavelet characterization of Besov type Morrey spaces under the action of semigroup. As an application, we obtain the well-posedness of smooth solution for the generalized Navier–Stokes equations with initial data in some critical homogeneous Besov type Morrey spaces (B˙p,pγ1,γ2)n (12<β<1, γ1−γ2=1−2β), 1<p⩽2 and np+2β−2<γ2<np or 2<p<∞, and max{np+2β−2,β−1}<γ2<np, with divergence free. These critical homogeneous Besov type Morrey spaces are larger than corresponding classical Besov spaces and cover Q spaces.