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abstract: 'Considering initial data in $\dot{H}^s$, with $\frac{1}{2} < s < \frac{3}{2}$, this paper is devoted to the study of possible blowing-up Navier-Stokes solutions such that ${\displaystyle}{({T_{*}}(u_{0}) -t)^{\frac{1}{2} (s- \frac{1}{2})} \,\, \| u \|_{\dot{H}^s} }$ 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.'
address: '([[Eugénie Poulon]{}]{}) Laboratoire Jacques-Louis Lions - UMR 7598, Université Pierre et Marie Curie, Boîte courrier 187, 4 place Jussieu, 75252 Paris Cedex 05, France'
author:
- Eugénie Poulon
title: 'About the possibility of minimal blow up for Navier-Stokes solutions with data in $\dot{H}^s(\R^3)$'
---

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Introduction and statement of main result
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We consider the Navier-Stokes system for incompressible viscous fluids evolving in the whole space $\R^{3}$. Denoting by $u$ the velocity, a vector field in $\R^3$, by $p$ in $\R$ the pressure function, the Cauchy problem for the homogeneous incompressible Navier-Stokes system is given by $$\left \lbrace \begin {array}{ccc}  \partial_{t}u + u\cdot\nabla{u}-\Delta{u}&=&-\nabla{p}\\ {\mathop{\rm div}\nolimits}u&=&0\\
 u_{|t=0}&=&u_{0}.\\ \end{array}
\right.$$ [0.2cm]{}

