On the refined analyticity radius of 3-D generalized Navier-Stokes equations

We analyze the instantaneous growth of analyticity radius for three dimensional generalized Navier-Stokes equations. For the subcritical $H^{\gamma }({\mathbb{R}}^3)$ case with $\gamma >\frac{1}{2},$ we prove that there exists a positive time $t_0$ so that for any $t\in ]0,t_0]$, the radius of analyticity of the solution $u$ satisfies the pointwise-in-time lower bound $$\mathrm{r}\mathrm{a}\mathrm{d}(u)(t)\ge \sqrt{(2\gamma -1)t(\mathrm{\mid }\ln t\mathrm{\mid }+\ln \mathrm{\mid }\ln t\mathrm{\mid }+K_t)},$$ where $K_t\to \mathrm{\infty }$ as $t\to 0^{+}$. This in particular gives a nontrivial improvement of the previous result by Herbst and Skibsted in \cite{HS} for the case $\gamma \in ]1\mathrm{/}2,3\mathrm{/}2[$ and also settles the decade-long open question in \cite{HS}, namely, whether or not ${\mathrm{lim\hspace{0.17em}inf}}_{t\to 0^{+}}\frac{\mathrm{r}\mathrm{a}\mathrm{d}(u)(t)}{\sqrt{t\mathrm{\mid }\ln t\mathrm{\mid }}}\ge \sqrt{2\gamma -1}$ for all $\gamma \ge \frac{3}{2}$. For the critical case $H^{\frac{1}{2}}({\mathbb{R}}^3)$, we prove that there exists $t_1>0$ so that for any $t\in ]0,t_1],$ $\mathrm{r}\mathrm{a}\mathrm{d}(u)(t)\ge \lambda (t)\sqrt{t}$ with $\lambda (t)$ satisfying ${\lim }_{t\to 0^{+}}\lambda (t)=\mathrm{\infty }\mathrm{.}$