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

TL;DR

This paper investigates the growth of the analyticity radius for solutions to 3D generalized Navier-Stokes equations, providing new bounds and settling open questions about its behavior near zero time.

Contribution

The paper establishes improved lower bounds on the analyticity radius for solutions in subcritical and critical Sobolev spaces, resolving longstanding open problems.

Findings

For $ ext{H}^eta$ with $eta>rac{1}{2}$, the analyticity radius grows at least as fast as $igl((2eta-1)t(| ext{ln} t|+ ext{ln}| ext{ln} t|+K_t)igr)^{1/2}$.

In the critical case $ ext{H}^{1/2}$, the analyticity radius exceeds $ ext{const} imes ext{sqrt}(t)$ with an unbounded coefficient as $t o 0^+$.

The results improve previous bounds and answer open questions about the precise asymptotic growth of analyticity radius.

Abstract

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 $\lim_{t\to 0^+}\lambda(t)=\infty.$