On the Radius of Analyticity of Solutions to 3D Navier-Stokes System with Initial Data in Lp*

Ruilin HUPing ZHANG

1Academy of Mathematics &Systems Science,the Chinese Academy of Sciences,Beijing 100190,China.E-mail:huruilin16@mails.ucas.an.cn

2Academy of Mathematics &Systems Science and Hua Loo-Keng Key Laboratory of Mathematics,the Chinese Academy of Sciences,Beijing 100190,China;School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China.E-mail:zp@amss.ac.cn

Abstract Given initial datafor somethe auhtors first prove that 3D incompressible Navier-Stokes system has a unique solution u=uL+v withand for some positive time T.Then they derive an explicit lower bound for the radius of space analyticity of v,which in particular extends the corresponding results in[Chemin,J.-Y.,Gallagher,I.and Zhang,P.,On the radius of analyticity of solutions to semi-linear parabolic system,Math.Res.Lett.,27,2020,1631–1643,Herbst,I.and Skibsted,E.,Analyticity estimates for the Navier-Stokes equations,Adv.in Math.,228,2011,1990–2033]with initial data in.

Keywords Incompressible Navier-Stokes equations,Radius of analyticity,Littlewood-Paley theory

1 Introduction

In the rest of this paper,we shall always use the convention that:For a ?b,we mean that there is a uniform constant C,which may be different on different lines,such that a ≤Cb.

Let us complete this section by the sketch of this paper.

In Section 2,we shall present the proof of Theorem 1.1.Section 3 is devoted to the proof of Theorem 1.2.In Section 4,we shall present the proof of Theorem 1.3.Finally in the Appendix,we shall collect some basic facts on Littlewood-Paley theory.

2 The Sobolev Regularity of v

3 The Radius of Analyticity of v for

4 The Critical Case p = 3

Appendix A Tool Box on Littlewood-Paley Theory