Large time behaviour of solutions to the 3D-NSE in $\mathcal X^{\sigma}$ spaces

TL;DR

This paper investigates the long-term decay behavior of solutions to the 3D Navier-Stokes equations within specific function spaces, revealing decay rates depending on the regularity index and employing Fourier analysis techniques.

Contribution

It establishes decay rates for solutions in $ ext{X}^\sigma$ spaces for $\sigma > -3/2$, extending understanding of solution behavior over time in these function spaces.

Findings

Solutions decay at least as fast as $t^{-(\sigma+3/2)/2}$ for $\sigma > -3/2$

Decay estimates are derived using Fourier analysis and standard PDE techniques

Results apply to solutions in the space $C( ext{R}^+, L^2 igcap ext{X}^{-1})$

Abstract

In this paper we study the incompressible Navier-Stokes equations in $L^2(\mathbb R^3)\cap\mathcal X^{-1}(\mathbb R^3)$. In the global existence case, we establish that if the solution $u$ is in the space $C(\mathbb R^+,L^2\cap\mathcal X^{-1})$, then for $\sigma>-3/2$ the decay of $\|u(t)\|_{\mathcal X^\sigma}$ is at least of the order of $t^{-\frac{\sigma+\frac{3}{2}}{2}}$. Fourier analysis and standard techniques are used.

Equations

$$(NS)\{\begin{array}{l}{\partial }_{t}u-\nu \Delta u+u.\nabla u=-\nabla pin{\mathbb{R}}^{+}\times {\mathbb{R}}^3\\ \mathrm{div}u=0\; in{\mathbb{R}}^{+}\times {\mathbb{R}}^3\\ u(0,x)=u^0(x)in{\mathbb{R}}^3,\end{array}$$

$$\Vert f_{\lambda }{\Vert }_X=\Vert f{\Vert }_X;\forall \lambda >0,\forall f\in X,$$

$$f_{\lambda }(x)=\lambda f(\lambda x).$$

$$\begin{array}[]{lcl}\|f_{\lambda}\|_{X}&=&\lambda^{\alpha}\|f\|_{X},\;\forall(\lambda,f)\in(0,\infty)\times X\\ \|g_{\lambda}\|_{Y}&=&\lambda^{\alpha}\|g\|_{Y},\;\forall(\lambda,g)\in(0,\infty)\times Y.\end{array}$$

$$\begin{array}[]{lcl}\dot{H}^{s}(\mathbb{R}^{3})&\approx&L^{p}(\mathbb{R}^{3}),\;\frac{1}{p}+\frac{s}{3}=\frac{1}{2},\;0\leq s<3/2\\ \mathcal{X}^{\sigma}(\mathbb{R}^{3})&\approx&\dot{H}^{\sigma+\frac{3}{2}}(\mathbb{R}^{3}),\;\forall\sigma\in\mathbb{R}.\end{array}$$

$$u(t)=e^{\nu t\Delta }{u}^0-{\int }_0^t{e}^{\nu (t-z)\Delta }\mathbb{P}(u.\nabla u)dz$$

$$\begin{array}[]{lcl}\|u(t)\|_{L^{2}}&\leq&\displaystyle\|e^{\nu t\Delta}u^{0}\|_{L^{2}}+\int_{0}^{t}\|e^{\nu(t-z)\Delta}\mathbb{P}(u.\nabla u)\|_{L^{2}}dz\\ &\leq&\displaystyle\|u^{0}\|_{L^{2}}+\int_{0}^{t}\|u\nabla u\|_{L^{2}}dz\\ &\leq&\displaystyle\|u^{0}\|_{L^{2}}+\int_{0}^{t}\|u\|_{L^{2}}\|\nabla u\|_{L^{\infty}}dz.\end{array}$$

$$\displaystyle\|u(t)\|_{L^{2}}\leq\|u^{0}\|_{L^{2}}\exp\big{(}(2\pi)^{-3}\int_{0}^{t}\|u\|_{\mathcal{X}^{1}}\big{)}.$$

$$\Vert u(t){\Vert }_{{\mathcal{X}}^{-1}}+\frac{\nu }{2}{\int }_0^t\Vert u{\Vert }_{{\mathcal{X}}^1}\le \Vert u^0{\Vert }_{{\mathcal{X}}^{-1}};\forall t\ge 0.$$

$$\begin{array}[]{lcl}\|u(t)\|_{L^{2}}&\leq&\displaystyle\|u(t_{0})\|_{L^{2}}\exp\big{(}(2\pi)^{-3}\int_{t_{0}}^{\infty}\|u(z)\|_{\mathcal{X}^{1}}dz\big{)}\\ &\leq&\displaystyle\|u(t_{0})\|_{L^{2}}\exp\big{(}(2\pi)^{-3}\frac{2\|u(t_{0})\|_{\mathcal{X}^{-1}}}{\nu}\big{)}\\ &\leq&\displaystyle\|u(t_{0})\|_{L^{2}}\exp\big{(}(2\pi)^{-3}\big{)}\\ &\leq&2\|u(t_{0})\|_{L^{2}},\end{array}$$

$$\Vert u(t)\Vert \le 2\underset{0\le z\le t_0}{\max }\Vert u(z){\Vert }_{L^2},\forall t\ge 0.$$

$$\Vert u(t){\Vert }_{L^2}\le 2\Vert u^0{\Vert }_{L^2},\forall t\ge 0.$$

$$\underset{t\to \infty }{\lim }\Vert u(t){\Vert }_{{\mathcal{X}}^{-1}}=0.$$

$$\Vert \exp (\sqrt{\nu t}|D|)u(t){\Vert }_{{\mathcal{X}}^{-1}}+\frac{\nu }{2}{\int }_0^t\Vert \exp (\sqrt{\nu z}|D|)u(z){\Vert }_{{\mathcal{X}}^1}dz\le 2\Vert u^0{\Vert }_{{\mathcal{X}}^{-1}},\forall t\ge 0.$$

$$\underset{t\to +\infty }{\lim }\Vert u(t){\Vert }_{{\mathcal{X}}^{-1}\cap L^2}=0,$$

$$\Vert u(t){\Vert }_{{\mathcal{X}}^{-1}}=o(t^{-\frac{1}{4}});t\to +\infty .$$

$$\Vert u(t){\Vert }_{{\mathcal{X}}^{\sigma }}=o(t^{-\frac{\sigma +3/2}{2}});t\to +\infty .$$

$$\mathcal{F}(f)(\xi )=\hat{f}(\xi )={\int }_{{\mathbb{R}}^3}\exp (-ix.\xi )f(x)dx,\xi =({\xi }_1,{\xi }_2,{\xi }_3)\in {\mathbb{R}}^3.$$

$${\mathcal{F}}^{-1}(g)(x)=(2\pi {)}^{-3}{\int }_{{\mathbb{R}}^3}\exp (i\xi .x)g(\xi )d\xi ,x=(x_1,x_2,x_3)\in {\mathbb{R}}^3.$$

$$(f\ast g)(x):={\int }_{{\mathbb{R}}^3}f(y)g(x-y)dy.$$

$$f\otimes g:=(g_{1}f,g_{2}f,g_{3}f),$$

$$\mathrm{div}(f\otimes g):=(\mathrm{div}(g_{1}f),\mathrm{div}(g_{2}f),\mathrm{div}(g_{3}f)).$$

$$\mathrm{div}(f\otimes g):=g_1{\partial }_{1}f+g_2{\partial }_{2}f+g_3{\partial }_{3}f:=g.\nabla f.$$

$$\Vert f{\Vert }_{{\mathcal{X}}^0({\mathbb{R}}^3)}\le \Vert f{\Vert }_{{\mathcal{X}}^{-1}({\mathbb{R}}^3)}^{1/2}\Vert f{\Vert }_{{\mathcal{X}}^1({\mathbb{R}}^3)}^{1/2},\forall f\in {\mathcal{X}}^{-1}({\mathbb{R}}^3)\cap {\mathcal{X}}^1({\mathbb{R}}^3).$$

$$\begin{array}[]{lcl}\|f\|_{\mathcal{X}^{0}}&=&\displaystyle\int_{\mathbb{R}^{3}}|\widehat{f}(\xi)|d\xi\\ &\leq&\displaystyle\int_{\mathbb{R}^{3}}|\xi|^{-1/2}|\widehat{f}(\xi)|^{1/2}\frac{|\widehat{f}(\xi)|^{1/2}}{|\xi|^{1/2}}d\xi.\end{array}$$

$$\Vert f{\Vert }_{{\mathcal{X}}^{\sigma }({\mathbb{R}}^3)}\le C\Vert f{\Vert }_{L^2({\mathbb{R}}^3)}^{1-\frac{\sigma +\frac{3}{2}}{s}}\Vert f{\Vert }_{{\dot{H}}^s({\mathbb{R}}^3)}^{\frac{\sigma +\frac{3}{2}}{s}},\forall f\in H^s({\mathbb{R}}^3).$$

$$\Vert f{\Vert }_{{\mathcal{X}}^{\sigma }}=I_{\lambda }+J_{\lambda },$$

$$\begin{array}[]{lcl}I_{\lambda}&=&\int_{|\xi|<\lambda}|\xi|^{\sigma}|\widehat{f}(\xi)|d\xi\\ J_{\lambda}&=&\int_{|\xi|>\lambda}|\xi|^{\sigma}|\widehat{f}(\xi)|d\xi.\end{array}$$

$$\begin{array}[]{lcl}I_{\lambda}&\leq&\Big{(}\int_{|\xi|<\lambda}|\xi|^{2\sigma}d\xi\Big{)}^{1/2}\|f\|_{L^{2}}\\ &\leq&\frac{c}{\sqrt{2\sigma+3}}\lambda^{\sigma+\frac{3}{2}}\|f\|_{L^{2}}\end{array}$$

$$\begin{array}[]{lcl}J_{\lambda}&\leq&\Big{(}\int_{|\xi|>\lambda}|\xi|^{2(\sigma-s)}d\xi\Big{)}^{1/2}\|f\|_{\dot{H}^{s}}\\ &\leq&\Big{(}\int_{|\xi|>\lambda}|\xi|^{2(\sigma-s)}d\xi\Big{)}^{1/2}\|f\|_{\dot{H}^{s}}\\ &\leq&\frac{C}{\sqrt{s-\sigma-\frac{3}{2}}}\lambda^{\sigma+\frac{3}{2}-s}\|f\|_{\dot{H}^{s}}.\end{array}$$

Full text

1. 1 Introduction

2. 2 Notations and preliminary results

3. 2.1 Notations

4. 2.2 Preliminary results

5. 3 Well posedness results in $L^{2}(\mathbb{R}^{3})\cap\mathcal{X}^{-1}(\mathbb{R}^{3})$

6. 4 **Proof of Theorem **1.6

7. 5 **Long time decay in **$\mathcal{X}^{\sigma}$

1. Introduction

The $3D$ incompressible Navier-Stokes equations are given by:

[TABLE]

where $\nu>0$ is the viscosity of fluid, $u=u(t,x)=(u_{1},u_{2},u_{3})$ and $p=p(t,x)$ denote respectively the unknown velocity and the unknown pressure of the fluid at the point $(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3}$, and $(u.\nabla u):=u_{1}\partial_{1}u+u_{2}\partial_{2}u+u_{3}\partial_{3}u$, while $u^{0}=(u_{1}^{o}(x),u_{2}^{o}(x),u_{3}^{o}(x))$ is an initial given velocity. If $u^{0}$ is quite regular, the divergence free condition determines the pressure $p$.

The Navier-Stokes system has the following scaling property : If $u=u(x,t)$ is a solution of $(NS)$ with initial date $u^{0}=u^{0}(x)$ on the interval $[0,T]$, then for all $\lambda>0,u_{\lambda}=\lambda u(\lambda^{2}t,\lambda x)$ is a solution of $(NS)$ with initial date $u_{\lambda}(0,x)=\lambda u^{0}(\lambda x)$ on the interval $[0,\frac{T}{\lambda^{2}}].$

A fonctional space $(X,\|.\|_{X})$ is called critical space of $(NS)$ system if

[TABLE]

where

[TABLE]

Particularly, $L^{3}(\mathbb{R}^{3})$, $\dot{H}^{1/2}(\mathbb{R}^{3})$ and $\mathcal{X}^{-1}(\mathbb{R}^{3})$ are critical spaces for the system $(NS)$. In order to explain the idea of studying the $(NS)$ system in the space $L^{2}\cap\mathcal{X}^{-1}$, we introduce the following notation : Two functional spaces $(X,\|.\|_{X})$ and $(Y,\|.\|_{Y})$ are called ”have the same scaling” if, there is a real number $\alpha$ such that

[TABLE]

In this case we note $X\approx Y$. For example:

[TABLE]

The second is a counter-example of two functional spaces that have the same scaling and are not comparable (see [4] for $\dot{H}^{1/2}(\mathbb{R}^{3})$ and $\mathcal{X}^{-1}(\mathbb{R}^{3})$). Now, We are ready to give the motivation for this work : Inspired by the works [9], [3] and [7] where they proved the decay results of a global solution of $(NS)$ in homogeneous Sobolev spaces by starting from the $H^{s}=L^{2}\cap\dot{H}^{s}$ solutions. Here we study the Navier-Stokes system $(NS)$ starting from the $L^{2}\cap\mathcal{X}^{-1}$ solutions and proving some optimal decay results. Our first result is the following.

Theorem 1.1**.**

Let $u^{0}\in\mathcal{X}^{-1}(\mathbb{R}^{3})\cap L^{2}(\mathbb{R}^{3})$ be a divergence free vector fields, then there is a time $T>0$ and unique solution $u\in C([0,T],\mathcal{X}^{-1}(\mathbb{R}^{3})\cap L^{2}(\mathbb{R}^{3}))$. Moreover $u\in L^{1}([0,T],\mathcal{X}^{1}(\mathbb{R}^{3}))$. If $\|u^{0}\|_{\mathcal{X}^{-1}}<\nu$, then $u$ is global.

Remark 1.2*.*

(i) If the maximal time $T^{*}$ is finite then $\int_{0}^{T^{*}}\|u(t)\|_{\mathcal{X}^{1}}=+\infty.$ Indeed : The integral form of the system $(NS)$:

[TABLE]

implies

[TABLE]

Using the fact $\|\nabla u\|_{L^{\infty}}\leq(2\pi)^{-3}\|u\|_{\mathcal{X}^{1}}$ and Gronwall lemma we get

[TABLE]

Then, if $\int_{0}^{T^{*}}\|u\|_{\mathcal{X}^{1}}$ is finite we get $u\in C([0,T^{*}),L^{2}\cap\mathcal{X}^{-1})\cap L^{\infty}([0,T^{*}),L^{2}\cap\mathcal{X}^{-1})$. Then the solution lives beyond the time $T^{*}$ which contradicts the fact that $T^{*}$ is the maximum time of existence.

(ii) If $\|u^{0}\|_{\mathcal{X}^{-1}}<\nu/2$, the above remark and [4] imply the global existence of solution $u$ of $(NS)$ with $u\in C(\mathbb{R}^{+},\mathcal{X}^{-1})\cap L^{1}(\mathbb{R}^{+},\mathcal{X}^{1})\cap C(\mathbb{R}^{+},L^{2})$. Moreover,

[TABLE]

(iii) Using (i)-(ii) and [4], we get if $u\in C(\mathbb{R}^{+},L^{2}\cap\mathcal{X}^{-1})$ is a global solution of $(NS)$, then $u\in L^{1}(\mathbb{R}^{+},\mathcal{X}^{1}(\mathbb{R}^{3}))$.

(iv) Using (i)-(ii)-(iii) and [4], we get if $u\in C(\mathbb{R}^{+},L^{2}\cap\mathcal{X}^{-1})$ is a global solution of $(NS)$, then $u\in\mathcal{C}_{b}(\mathbb{R}^{+},L^{2}(\mathbb{R}^{3}))$. Indeed: By [4] there is a time $t_{0}\geq 0$ such that $\|u(t_{0})\|_{\mathcal{X}^{-1}}<\nu/2$. Then (i)-(ii) imply for $t\geq t_{0}$

[TABLE]

which implies

[TABLE]

Particularly, if $\|u^{0}\|_{\mathcal{X}^{-1}}<\nu/2$ we get

[TABLE]

Before stating the result of decay of the global solution of $(NS)$, we recall the following results which will be useful in the following

Theorem 1.3**.**

(see [4]) Let $u\in C(\mathbb{R}^{+},\mathcal{X}^{-1}(\mathbb{R}^{3}))$ be a global solution of Navier-Stokes system. Then

[TABLE]

Theorem 1.4**.**

(see [2]) There exists a positive constant $\epsilon_{0}>0$ such that for any initial data $u^{0}$ in $\mathcal{X}^{-1}(\mathbb{R}^{3})$ with $\|u^{0}\|_{\mathcal{X}^{-1}}<\epsilon_{0}$, the solution of Navier-Stokes system is analytic in the sense that

[TABLE]

Theorem 1.5**.**

(see [6]) For any initial data $u^{0}\in H^{s}(\mathbb{R}^{3})$ with $\rm{div}\,u^{0}=0$, there exists a unique solution $u\in C([0,T_{0}],H^{s}(\mathbb{R}^{3}))$ such that $T_{0}=T_{0}(s,\|u^{0}\|_{H^{s}}).$

Our second result is the following.

Theorem 1.6**.**

Let $u\in{\mathcal{C}}(\mathbb{R}^{+},\mathcal{X}^{-1}(\mathbb{R}^{3})\cap L^{2}(\mathbb{R}^{3}))$ be a global solution of Navier-Stokes system. Then

[TABLE]

Precisely,

[TABLE]

Using theorem 1.6 and theorem 1.4 which characterizes the regularizing effect of the Navier-Stokes equations, we get the following decay result of $\|u(t)\|_{\mathcal{X}^{\sigma}}$.

Corollary 1.7**.**

Let $u\in{\mathcal{C}}(\mathbb{R}^{+},\mathcal{X}^{-1}(\mathbb{R}^{3})\cap L^{2}(\mathbb{R}^{3}))$ be a global solution of Navier-Stokes system. Then, for all $\sigma>-3/2$, we have $u\in{\mathcal{C}}((0,+\infty),\mathcal{X}^{\sigma})$ and

[TABLE]

The remainder of our paper is organized as follows. In the second section we give some notations, definitions and preliminary results. Section 3 is devoted to prove the well posedness of $(NS)$ in $L^{2}\cap\mathcal{X}^{-1}$ space, this proof used the Fixed Point Theorem with a good choice of space $X=\mathcal{C}([0,T],L^{2}\cap\mathcal{X}^{-1})\cap L^{1}([0,T],\mathcal{X}^{1})$. In section 4 we prove the decay of global solutions in $L^{2}\cap\mathcal{X}^{-1}$, this proof used a Fourier analysis and standard techniques. Section 5 is devoted to prove the decay results of the global solution in $\mathcal{X}^{\sigma}$, this proof uses in a fundamental way the decay in $L^{2}\cap\mathcal{X}^{-1}$.

2. Notations and preliminary results

2.1. Notations

In this section, we collect some notations and definitions that will be used later.

$\bullet$ The Fourier transformation is normalized as

[TABLE]

$\bullet$ The inverse Fourier formula is

[TABLE]

$\bullet$ The convolution product of a suitable pair of function $f$ and $g$ on $\mathbb{R}^{3}$ is given by

[TABLE]

$\bullet$ If $f=(f_{1},f_{2},f_{3})$ and $g=(g_{1},g_{2},g_{3})$ are two vector fields, we set

[TABLE]

and

[TABLE]

Moreover, if $\rm{div}\,g=0$ we obtain

[TABLE]

$\bullet$ Let $(B,||.||)$, be a Banach space, $1\leq p\leq\infty$ and $T>0$. We define $L^{p}_{T}(B)$ the space of all measurable functions $[0,t]\ni t\mapsto f(t)\in B$ such that $t\mapsto||f(t)||\in L^{p}([0,T])$.

$\bullet$ The Sobolev space $H^{s}(\mathbb{R}^{3})=\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{3});\;(1+|\xi|^{2})^{s/2}\widehat{f}\in L^{2}(\mathbb{R}^{3})\}$.

$\bullet$ The homogeneous Sobolev space $\dot{H}^{s}(\mathbb{R}^{3})=\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{3});\;\widehat{f}\in L^{1}_{loc}\;{\rm and}\;|\xi|^{s}\widehat{f}\in L^{2}(\mathbb{R}^{3})\}$.

$\bullet$ The Lei-Lin space $\mathcal{X}^{\sigma}(\mathbb{R}^{3})=\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{3});\;\widehat{f}\in L^{1}_{loc}\;{\rm and}\;|\xi|^{\sigma}\widehat{f}\in L^{1}(\mathbb{R}^{3})\}$.

2.2. Preliminary results

In this section, we recall some classical results and we give new technical lemmas.

Lemma 2.1**.**

We have $\mathcal{X}^{-1}(\mathbb{R}^{3})\cap\mathcal{X}^{1}(\mathbb{R}^{3})\hookrightarrow\mathcal{X}^{0}(\mathbb{R}^{3})$. Precisely, we have

[TABLE]

Proof. We can write

[TABLE]

Cauchy-Schwartz inequality gives the result.

Lemma 2.2**.**

Let $\sigma,s\in\mathbb{R}$ such that $0<\sigma+\frac{3}{2}<s$. Then $H^{s}(\mathbb{R}^{3})\hookrightarrow\mathcal{X}^{\sigma}(\mathbb{R}^{3})$. Precisely, there is a constant $C=C(s,\sigma)$ such that

[TABLE]

Proof. For $\lambda>0$, we have

[TABLE]

with

[TABLE]

We have

[TABLE]

[TABLE]

For $\lambda=(\|f\|_{\dot{H}^{s}}/\|f\|_{L^{2}})^{1/s}$, we obtain the desired result.

Lemma 2.3**.**

Let $\sigma_{0}>-3/2.$ If we have

[TABLE]

Precisely

[TABLE]

Proof. For $\lambda>0$, we have

[TABLE]

with

[TABLE]

We have

[TABLE]

which imply

[TABLE]

For $\lambda=(\|f\|_{{\mathcal{X}^{\sigma_{0}}}}/\|f\|_{L^{2}})^{1/(3/2+\sigma_{0})}$, we obtain

[TABLE]

Lemma 2.4**.**

Let $f,g\in L^{\infty}_{T}(\mathcal{X}^{-1}(\mathbb{R}^{3})\cap L^{2}(\mathbb{R}^{3}))\cap L^{1}_{T}(\mathcal{X}^{1}(\mathbb{R}^{3}))$ such that ${\rm{div}}\,f=0$ almost everywhere. Then

[TABLE]

[TABLE]

[TABLE]

**Proof.

$\bullet$** Proof of (2.4): We can write

[TABLE]

$\bullet$ Proof of (2.5): We can write

[TABLE]

$\bullet$ Proof of (2.6): We can write

[TABLE]

Lemma 2.5**.**

Let $T>0$ and $f:[0,T]\rightarrow\mathbb{R}_{+}$ be continuous function such that

[TABLE]

with $M_{0}\geq 0$ and $\theta_{1},\theta_{2}\in(0,1).$ Then

[TABLE]

Proof. As $f$ is a positive and continuous function, then there is a time $t_{0}\in[0,T]$ such that

[TABLE]

Applying (2.7) at $t=t_{0}$ we get

[TABLE]

which implies $f(t_{0})\leq\frac{M_{0}}{1-\theta_{1}}.$ As $f(t_{0})=\max_{0\leq t\leq T}f(t)$, we get the desired result.

Remark 2.6*.*

Applying Lemma 2.5 to a positive continuous function $f:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ satisfying

[TABLE]

with $M_{0}\geq 0$ and $\theta_{1},\theta_{2}\in(0,1)$, we obtain

[TABLE]

3. Well posedness results in $L^{2}(\mathbb{R}^{3})\cap\mathcal{X}^{-1}(\mathbb{R}^{3})$

In this section we prove Theorem 1.1. To prove the existence result we need the following remark : For $f\in L^{2}(\mathbb{R}^{3})\cap\mathcal{X}^{-1}(\mathbb{R}^{3})$ and $\varepsilon_{0}>0$ there is $\lambda>0$ such that

[TABLE]

Precisely, just take $\lambda=\frac{\varepsilon_{0}^{2}}{4\|f\|_{L^{2}}^{2}+1}$. Then we can choose $\lambda_{0}>0$ such that

[TABLE]

Consider then the Navier-Stokes system

[TABLE]

If the system $(NS_{\lambda_{0}})$ has a unique solution $v$ in $C([0,T],L^{2}\cap\mathcal{X}^{-1})$, then $u=\lambda_{0}^{-1}v(\lambda_{0}^{-2}t,\lambda_{0}^{-1}x)$ is a solution of Navier-Stokes system starting by $u^{0}$. Therefore, we can assume in the following that

[TABLE]

Let’s go back to the proof of Theorem 1.1. A uniqueness in $L^{2}\cap\mathcal{X}^{-1}$ is given by the uniqueness in $\mathcal{X}^{-1}$,(see [8]). It remains a proven existence, for this let $k_{0}\in\mathbb{N}^{*}$ such that

[TABLE]

Put

[TABLE]

We have $a^{0}\in H^{s}(\mathbb{R}^{3})$,(for all $s\geq 0$) and

[TABLE]

[TABLE]

Moreover

[TABLE]

There is a time $T_{0}>0$ such that the system $(NS)$ has a unique solution $a$ in $C([0,T_{0}],H^{4}(\mathbb{R}^{3}))$ with initial condition $a^{0}$ (see [6]). Using the fact(see Lemma 2.2)

[TABLE]

we get

[TABLE]

Using the regularity of the function $a$ and inequality (3.4), we obtain

[TABLE]

Put $b=u-a$, $b$ satisfies the following system

[TABLE]

The integral form of $(RNS)$ is

[TABLE]

Put

[TABLE]

For $T>0$ put the space

[TABLE]

This vector space is equipped with the norm

[TABLE]

For $\varepsilon,T>0$ (to fixed later), such that $T\leq T_{0}$, put the closed subset of $X_{T}$ defined by

[TABLE]

Explanation of the choice of $\varepsilon$ and $T$ : We have

[TABLE]

Dominate Convergence Theorem implies

[TABLE]

Let $0<\varepsilon<1/24$ and $0<T\leq T_{0}$ such that

[TABLE]

These choices are possible just use the equations (3.3)-(3.5)-(3.7). Now we want to prepare to apply the Fixed Point Theorem, for this we prove the following

[TABLE]

[TABLE]

Proof of (3.8):** Using inequality (2.4), we obtain**

[TABLE]

Then

[TABLE]

Similarly, inequality (2.5) gives

[TABLE]

Then

[TABLE]

Finally, inequality (2.6) gives

[TABLE]

Then,

[TABLE]

**Therefore inequalities (3.13)-(3.11)-(3.12) imply (3.8).

Proof of (3.9): Using inequality (2.4), we obtain**

[TABLE]

Then

[TABLE]

Similarly, inequality (2.5) gives

[TABLE]

Then

[TABLE]

Finally, inequality (2.6) gives

[TABLE]

Then,

[TABLE]

Therefore inequalities (3.13)-(3.11)-(3.12) gives

[TABLE]

Fixed Point Theorem gives the existence and uniqueness of solution of $(RNS)$ in $C_{T}(L^{2}\cap\mathcal{X}^{-1})\cap L^{1}_{T}(\mathcal{X}^{1}).$ Therefore, we can deduce the existence and uniqueness of a local solution for Navier-Stokes system.

4. Proof of Theorem 1.6

Proof of (1.3) :** In this subsection we want to prove the long time decay in $L^{2}\cap\mathcal{X}^{-1}$. Let $u\in C(\mathbb{R}^{+},L^{2}\cap\mathcal{X}^{-1})$ be global solution of $(NS)$. By [4] we have**

[TABLE]

Now, prove that $\limsup_{t\rightarrow\infty}\|u(t)\|_{L^{2}}=0.$ For a strictly positive real number $\delta$ and a given distribution $f$, we define the operators $A_{\delta}(D)$ and $B_{\delta}(D)$, respectively, by the following:

[TABLE]

Let $u$ be a solution of $(NS)$. Denote by $w_{\delta}=A_{\delta}(D)u$ and $v_{\delta}=B_{\delta}(D)u$, respectively, the low-frequency part and the high-frequency part of $u$ and so on $w_{\delta}^{0}$ and $v_{\delta}^{0}$ for the initial data $u^{0}$. Applying the pseudo-differential operator $A_{\delta}(D)$ to the $(NS)$, we get

[TABLE]

Taking the $L^{2}(\mathbb{R}^{3})$-inner product and using the fact $A_{\delta}(D)^{2}=A_{\delta}(D)$, we obtain

[TABLE]

Integrating with respect to time and using Remark 1.2-(iv), we obtain

[TABLE]

where $m_{0}=(2\pi)^{-3}\|u\|_{L^{\infty}(\mathbb{R}^{+},L^{2})}$. Also using Remark 1.2-(iii) we get $\|w_{\delta}(t)\|^{2}_{L^{2}}\leq M_{\delta},$ where

[TABLE]

On the one hand, it is clear that $\lim_{\delta\rightarrow 0}\|w_{\delta}^{0}\|^{2}_{L^{2}}=0.$ On the other, we have $\lim_{\delta\rightarrow 0}\|w_{\delta(t)}\|_{\mathcal{X}^{1}}=0$ and $\|w_{\delta(t)}\|_{\mathcal{X}^{1}}\leq\|u(t)\|_{\mathcal{X}^{1}}\in L^{1}([0,\infty)).$ Then Dominate Convergence Theorem implies that

[TABLE]

Hence, $\lim_{\delta\rightarrow 0}M_{\delta}=0$ and thus

[TABLE]

Let us investigate the high-frequency part. To do so, one applies the pseudo-differential operator $B_{\delta}(D)$ to the $(NS)$ to get

[TABLE]

The integral form of $v_{\delta}$ is

[TABLE]

Taking the $L^{2}(\mathbb{R}^{3})$ norm, we obtain

[TABLE]

Then

[TABLE]

We have

[TABLE]

This leads to the fact that the function $(t\rightarrow\|v_{\delta}(t)\|_{L^{2}})$ is both continuous and Lebesgue integrable over $\mathbb{R}^{+}$. Let $\varepsilon>0$ be positive real number small enough. Firstly, equation (4.2 ) implies that some $\delta_{\varepsilon}>0$ exists such that

[TABLE]

Secondly, consider the set $R_{\delta_{\varepsilon}}$ defined by

[TABLE]

If we denote by $\lambda_{1}(R_{\delta_{\varepsilon}})$ the Lebesgue measure of $R_{\delta_{\varepsilon}}$, we have

[TABLE]

By this, we can deduce that $\lambda_{1}(R_{\delta_{\varepsilon}})\leq T_{\varepsilon},$ where $T_{\varepsilon}=(2/\varepsilon)\int_{0}^{\infty}\|v_{\delta_{0}}(t)\|_{L^{2}{(\mathbb{R}^{3})}}dt.$ Then, there is $t_{\varepsilon}\in[0,T_{\varepsilon}+1]$ such that $t_{\varepsilon}$ does not belong to $R_{\delta_{\varepsilon}}.$ This implies that

[TABLE]

Equations (4.4) and (4.6) together with triangular inequality imply that $\|u(t_{\varepsilon})\|_{L^{2}(\mathbb{R}^{3})}<\varepsilon.$ For $t\geq t_{\varepsilon}$, we have

[TABLE]

It suffices to replace $\varepsilon$ by $\varepsilon\exp(-(2\pi)^{-3}\int_{0}^{\infty}\|u\|_{\mathcal{X}^{1}})$ in (4.4)-(4.5)-(4.6) we get the desired result.

Proof of (1.4) :** In this subsection we want to give a precision for the decay of $\|u(t)\|_{\mathcal{X}^{-1}}$ at $\infty$. Let $\varepsilon>0$ such that $\varepsilon<\epsilon_{0}$($\epsilon_{0}$ is given by Theorem 1.4), by (1.3) we can suppose that,**

[TABLE]

Then, by Remark1.2-(ii)-(iv) we get $\|u(t)\|_{L^{2}}\leq 2\|u^{0}\|_{L^{2}}<\varepsilon$ for all $t\geq 0$ and

[TABLE]

For $\lambda>0$ and $t>00$, we have

[TABLE]

with

[TABLE]

We have

[TABLE]

and

[TABLE]

For a fixed time $t>0$ the $v:(z,x)\rightarrow u(\frac{t}{2}+z,x)$ satisfies $\|v(0)\|_{\mathcal{X}^{-1}}<\epsilon_{0}$ and it is the unique global solution of the following system,

[TABLE]

By Theorem 1.4, we get

[TABLE]

or

[TABLE]

For $z=\frac{t}{2}$, we get $\int_{\xi}e^{\sqrt{\nu t/2}|\xi|}\frac{|\widehat{u}(t,\xi)|}{|\xi|}d\xi\leq\|u(t/2)\|_{\mathcal{X}^{-1}}$, which implies

[TABLE]

Then

[TABLE]

Multiplying this inequality by $t^{1/4}$

[TABLE]

and choose $\lambda>0$ such that

[TABLE]

we obtain

[TABLE]

with

[TABLE]

Applying Lemma 2.5 and Remark 2.6 with

[TABLE]

we get

[TABLE]

Applying this result to the solution of the following system, for $a\geq 0$

[TABLE]

we obtain

[TABLE]

Then the fact $\lim_{a\rightarrow+\infty}\|u(a)\|_{L^{2}}=0$ implies the desired result.

5. Long time decay in $\mathcal{X}^{\sigma}$

In this section we want to prove Corollary 1.7.

First case**   $:-3/2<\sigma<-1.$ For $\lambda>0$ and $t>0$, we have**

[TABLE]

[TABLE]

We have

[TABLE]

and

[TABLE]

We get $\|u(t)\|_{\mathcal{X}^{\sigma}}=A\lambda^{\sigma+3/2}+B\lambda^{\sigma+1}:=\varphi(\lambda)$, with

[TABLE]

The study of the function $\varphi$ gives

[TABLE]

then

[TABLE]

For $\lambda=\lambda_{0}$, we get

[TABLE]

Then

[TABLE]

Theorem 1.5 implies

[TABLE]

which gives the desired result.

Second case**    $:-1<\sigma.$ By Theorem 1.5 we can assume that $\|u^{0}\|_{\mathcal{X}^{-1}}<\epsilon_{0}$ and Theorem 1.4 gives,**

[TABLE]

with $C_{\nu}=\nu^{-\frac{\sigma+1}{2}}\sup_{z\geq 0}z^{\sigma+1}e^{-rz}.$ Combining this result with the fact $\|u(t/2)\|_{\mathcal{X}^{-1}}=o(t^{-1/4})$ we get the desired result.