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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?

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  • $\begingroup$ Incidentally, I am not sure if you are quoting Fujita-Kato correctly. The time $T^*$ in the local existence paper of Fujita-Kato is not necessarily a function merely of $\|u_0\|_{\dot{H}^{1/2}}$. For any $s > 1/2$ you can choose $T^*$ as a function of $\|u_0\|_{H^{s}}$, but this doesn't hold in the critical case. (See Chapter 7 of Lemarie-Rieusset's "The Navier Stokes Problem in the 21st Century". Especially Theorems 7.1, 7.3, and 7.4.) $\endgroup$ – Willie Wong Sep 6 '18 at 17:37
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$\dot{H}^{1/2}$ 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".

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