  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
$Q^{\beta,-1}_{\alpha, loc}(\mathbb{R}^{3}).$ This result applies to the usual
incompressible Navier-Stokes equations and deduces a known link.
