On the instantaneous radius of analyticity of $L^p$ solutions to 3D Navier–Stokes system

Abstract

In this paper, we first investigate the instantaneous radius of space analyticity for the solutions of 3D Navier–Stokes system with initial data in the Besov spaces $\dot{B}^s_{p,q}({\mathbb R}^3)$ for $p\in ]1,\infty [,$ $q\in [1,\infty ]$ and $s\in \bigl [-1+\frac{3}{p},\frac{3}{p}\bigr [.$ Then for initial data $u_0\in L^p({\mathbb R}^3)$ with p in ]3, 6[,  we prove that 3D Navier–Stokes system has a unique solution $u=u_L+v$ with and $v\in {\widetilde{L}^\infty _T\Big (\dot{B}^{1-\frac{3}{p}}_{p,\frac{p}{2}}\Big )}\cap {\widetilde{L}^1_T\Big (\dot{B}^{3-\frac{3}{p}}_{p,\frac{p}{2}}\Big )}$ for some positive time T. Furthermore, we derive an explicit lower bound for the radius of space analyticity of v,  which in particular extends the corresponding results in Hu and Zhang (Chin Ann Math Ser B 43:749–772, 2022) with initial data in $L^p({\mathbb R}^3)$ for $p\in [3, 18/5[.$

Appendix A: Tool box on Littlewood–Paley theory

For the convenience of readers, we shall collect some basic facts on Littlewood–Paley theory in this section. Let us first recall Bernstein Lemma from [2]:

Lemma A.1

Let ${{\mathcal {B}}}$ be a ball of ${\mathbb R}^3$, and ${{\mathcal {C}}}$ a ring of ${\mathbb R}^3$; let $1\le p_2\le p_1\le \infty .$ Then there holds:

$$\begin{aligned} \begin{aligned} \text{ if }\ \ \textrm{Supp}\widehat{a}\subset 2^j{{\mathcal {B}}}&\Rightarrow \Vert \partial _{x}^\alpha a\Vert _{L^{p_1}} \lesssim 2^{j\left( |\alpha |+3\left( \frac{1}{p_2}-\frac{1}{p_1}\right) \right) } \Vert a\Vert _{L^{p_2}};\\ \text{ if }\ \ \textrm{Supp}\widehat{a}\subset 2^j{{\mathcal {C}}}&\Rightarrow \Vert a\Vert _{L^{p_1}} \lesssim 2^{-jN}\sup _{|\alpha |=N} \Vert \partial _{x}^\alpha a\Vert _{L^{p_1}}. \end{aligned} \end{aligned}$$

In order to obtain a better description of the regularizing effect of the transport-diffusion equation, we need to use Chemin-Lerner type space $\widetilde{L}^{r}_T(\dot{B}^{s}_{p,q})$.

Definition A.1

Let $r\in [1,\,+\infty ]$ and $T_0,T\in [0,\,+\infty ]$. If $r<\infty ,$ we define $\widetilde{L}^{r}(T_0,T; \dot{B}^{s}_{p,q})$ as the completion of $C([T_0,T]; \,{{\mathcal {S}}}({\mathbb R}^3))$ by the norm

with the usual change if $r=\infty .$ In particular, when $T_0=0,$ we denote for simplicity.

We also recall the action of the heat semigroup on distribution with the Fourier transform of which is supported in an annulus.

Lemma A.2

(Lemma 2.4 of [2]) Let ${\mathcal {C}}$ be an annulus, then there exists constants c and C, such that for any $p\in [1,\infty ]$ and any couple of $(t,\lambda )$, there holds

$$\begin{aligned} Supp\ \hat{u}\subseteq \lambda {\mathcal {C}}\ \Rightarrow \ \Vert e^{t\Delta }u\Vert _{L^p}\le Ce^{-ct\lambda ^2}\Vert u\Vert _{L^p} \end{aligned}$$

(A.1)

To deal with the estimate of product of two distributions, we constantly use the following para-differential decomposition from [5]: for any functions $f,g\in {\mathcal {S}}'({\mathbb R}^3)$,

$$\begin{aligned} fg=T_fg+{T}_gf+R(f,g), \end{aligned}$$

(A.2)

where