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

TL;DR

This paper investigates the existence of self-similar blow-up solutions in a generalized Navier-Stokes system with fractional Laplacian, using renormalization techniques to establish bounds and fixed points.

Contribution

It introduces a renormalization framework for the fractional Navier-Stokes equations and proves the existence of self-similar solutions with unbounded norms in finite time.

Findings

Existence of renormalization fixed points for certain dimensions and fractional powers.

Construction of a-priori bounds invariant under renormalization.

Proof of non-trivial self-similar solutions with finite-time blow-up.

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