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 $\gamma>\frac12,$ 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 $${\mathrm{rad}}(u)(t)\ge \sqrt{(2\gamma-1)t\bigl(|\ln t|+\ln|\ln t|+K_t\bigr)},$$ where $K_t \to \infty$ 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 $\gamma\in ]1/2,3/2[$ and also settles the decade-long open question in \cite{HS}, namely, whether or not $\liminf_{t\to 0^+}\frac {{\mathrm{ rad}}(u)(t)}{\sqrt{t|\ln t|}}\ge \sqrt{2\gamma-1}$ for all $\gamma\ge \frac32$. For the critical case $H^{\frac 12}(\mathbb R^3)$, we prove that there exists $t_1>0$ so that for any $t\in ]0, t_1],$ ${\mathrm {rad}}(u)(t)\ge \lambda(t)\sqrt{t}$ with $\lambda(t)$ satisfying $\lim_{t\to 0^+}\lambda(t)=\infty.$