Space-time analyticity and refined analyticity radius of the Navier-Stokes equations in the critical Besov spaces
Cong Wang
TL;DR
This work proves joint space-time analyticity for solutions to the incompressible NSE with small initial data in the critical Besov spaces $\dot{B}_{p,q}^{3/p-1}$ by blending Gevrey-class estimates in space with iterative temporal-derivative techniques, anchored by sharp heat-kernel Besov bounds. The authors show that for such data, the mild solution satisfies $\|t^n e^{\sqrt{t}\Lambda} \partial_t^n u\|_{E_{p,q}} \le \rho^{-1} C^n n^n$ for all $n$, implying space-time analyticity and yielding time-decay rates for higher derivatives. A corollary provides instantaneous lower bounds on the space-time analyticity radius, and a refined radius result is achieved through the introduction of $E_{p,q}^{\varepsilon}(T)$ with a Gevrey weight, establishing $\liminf_{t\to 0^+} \frac{\text{rad}_S(u(t))}{\sqrt{t}} = \infty$. These results advance understanding of analyticity scales in NSE within critical Besov spaces and have implications for numerical methods and control strategies that rely on analytic regularity.
Abstract
In this paper, we establish the space-time analyticity of global solutions to the incompressible Navier-Stokes equations with small initial data in critical \emph{Besov} spaces $\dot B^{3/p-1}_{p,q}$. Time decay rates of higher order space-time joint derivatives and instantaneous lower bounds of the analyticity radius follow as straightforward consequences. The method employed combines Gevrey-class estimates with iterative derivative techniques. Furthermore, we obtain a logarithmic improvement in the lower bound for the spatial analyticity radius of solutions for arbitrary initial data in critical \emph{Besov} spaces.