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?