  We demonstrate that the problem of existence of Leray self-similar blow up
solutions in a generalized mild Navier-Stokes system with the fractional
Laplacian $(-\Delta)^{\gamma/2}$ can be stated as a fixed point problem for a
"renormalization" operator. We proceed to construct {\it a-priori} bounds, that
is a renormalization invariant precompact set in an appropriate weighted
$L^p$-space.
  As a consequence of a-priori bounds, we prove existence of renormalization
fixed points for $d \ge 2$ and $d<\gamma <2 d+2$, and existence of non-trivial
Leray self-similar mild solutions in $C^\infty([0,T),(H^k)^d \cap (L^p)^d)$,
$k>0, p \ge 2$, whose $(L^p)^d$-norm becomes unbounded in finite time $T$.
