Renormalization and a-priori bounds for Leray self-similar solutions to the generalized mild Navier-Stokes equations

Abstract

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 $d<\gamma <2d+2$, 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 $T$.