About the possibility of minimal blow up for Navier-Stokes solutions with data in ${\dot{H}}^s(R^3)$

Abstract

Considering initial data in ${\dot{H}}^s$, with $\frac{1}{2} \textless{} s \textless{} \frac{3}{2}$, this paper is devoted to the study of possible blowing-up Navier-Stokes solutions such that $(T*(u\_{0}) -t)^{\frac{1}{2} (s- \frac{1}{2})} \,\, \| u \|\_{\dot{H}^s}}$ is bounded. Our result is in the spirit of the tremendous works of L. Escauriaza, G. Seregin, and V. $\breve{\mathrm{S}}$ver$\acute{\mathrm{a}}$k and I. Gallagher, G. Koch, F. Planchon, where they proved there is no blowing-up solution which remain bounded in $L^3(R^3)$. The main idea is that if such blowing-up solutions exist, they satisfy critical properties.