Let $u\_0 \in \dot{H}^{1/2}(\mathbb{R}^3)$. The Fujita-Kato theorem gives rise to a local unique solution $(t,x) \mapsto u(t,x)$ to the Navier-Stokes equations $$\left\{
\begin{array}{ccc}
\partial \_t u + u\cdot \nabla u- \Delta u + \nabla p&=&0\\
div \;u&=&0 \\
u(t=0)&=&u\_0.
\end{array}
\right.$$
The Fujita-Kato asserts that there exists a time $T^\*=T(\|u\_0\|\_{\dot{H}^{1/2}})$ such that the solution $u$ is defined (and unique) on $[0,T^\*)$. Itis also known that there is a constant $c\_0$ such that if $\|u\_0\|\_{\dot H^{1/2}} < c\_0$ then $u$ is globally defined. 



> 
> Assume that $\|u\_0\|\_{\dot H^{1/2}} \gg c\_0$. Is there a constant $K=K(\|u\_0\|\_{\dot H^{1/2}})$ such
>  that if $T^\* > K$ then $T^\*= + \infty$? If yes, where can I read a
>  proof of it?
> 
> 
> 



A solution to the Navier-Stokes equation that is defined for on $[0,T]$ with $T$ large is global?
$\dot{H}^{1/2}$ is critical with respect to scaling. 


Let $\tilde{u}(t,x) = \lambda u(\lambda^2 t, \lambda x)$. Then $\tilde{u}$ solves the Navier-Stokes equation up to $\tilde{T}^\* = T^\* \lambda^{-2}$, with pressure $\tilde{p} = \lambda^2 p(\lambda^2 t, \lambda x)$. 


One can check that the corresponding initial data satisfies
$$ \|\tilde{u}\|\_{\dot{H}^{1/2}(\mathbb{R}^3)} = \|u\|\_{\dot{H}^{1/2}(\mathbb{R}^3)} $$


By letting $\lambda \searrow 0$ if you have a solution $u$ that exists up to time $1$, you will have a solution $\tilde{u}$ *with the same* $\dot{H}^{1/2}$ initial energy that exists for arbitrarily long time. 


This means that the statement you asked about is equivalent to global existence of Navier-Stokes for all initial data with finite $\dot{H}^{1/2}$ norm. Which tells you that what you are asking about is certainly not a "known result". 


