Mathematical Research Letters

Volume 27 (2020)

Number 6

On the radius of analyticity of solutions to semi-linear parabolic systems

Pages: 1631 – 1643



Jean-Yves Chemin (Laboratoire J.-L. Lions, Sorbonne Université, Paris, France)

Isabelle Gallagher (DMA, École normale supérieure, CNRS, PSL Research University, Paris, France; and UFR de mathématiques, Université de Paris, France)

Ping Zhang (Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China; and School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing, China)


We study the radius of analyticity $R(t)$ in space, of strong solutions to systems of scale-invariant semi-linear parabolic equations. It is well-known that near the initial time, $R(t)t^{-\frac{1}{2}}$ is bounded from below by a positive constant. In this paper we prove that $\displaystyle\liminf_{t\to 0} R(t)t^{-\frac{1}{2}}=\infty$, and assuming higher regularity for the initial data, we obtain an improved lower bound near time zero. As an application, we prove that for any global solution $u$ in $C([0,\infty); H^{\frac{1}{2}}(\mathbb{R}^3))$ of the Navier–Stokes equations, there holds $\displaystyle\liminf_{t\to \infty} R(t)t^{-\frac{1}{2}}= \infty$.

Part of the work was done when P. Zhang was visiting Laboratoire J. L. Lions of Sorbonne Université in the fall of 2018. He would like to thank the hospitality of the Laboratory. P. Zhang is partially supported by NSF of China under Grants 11731007 and 11688101, and innovation grant from National Center for Mathematics and Interdisciplinary Sciences. This work is also supported by the K.C. Wong Education Foundation.

Received 8 April 2020

Accepted 13 October 2020

Published 17 February 2021