Generalized Naiver-Stokes equations with initial data in local $Q$-type spaces

In this paper, we establish a link between Leray mollified solutions of the three-dimensional generalized Naiver-Stokes equations and mild solutions for initial data in the adherence of the test functions for the norm of $Q_{\alpha ,loc}^{\beta ,-1}({\mathbb{R}}^3)\mathrm{.}$ This result applies to the usual incompressible Navier-Stokes equations and deduces a known link.