Weak solutions for Navier--Stokes equations with initial data in weighted $L^2$ spaces

TL;DR

This paper proves the existence of global weak solutions to the 3D Navier-Stokes equations with initial data in weighted L2 spaces, expanding understanding of solutions with less regular initial conditions.

Contribution

It introduces new energy control methods to establish global weak solutions with initial data in weighted L2 spaces and provides a new proof for discretely self-similar solutions.

Findings

Existence of global weak solutions with initial data in weighted L2 spaces.

New energy control techniques for Navier-Stokes equations.

Alternative proof for discretely self-similar solutions.

Abstract

We show the existence of global weak solutions of the 3D Navier-Stokes equations with initial velocity in the weighted spaces L 2 w$\gamma$ , where w $\gamma$ (x) = (1 + |x|) --$\gamma$ and 0 < $\gamma$ $\le$ 2, using new energy controls. As application we give a new proof of the existence of global weak discretely self-similar solutions of the 3D Navier-Stokes equations for discretely self-similar initial velocities which are locally square inte-grable.

Equations

$$w_{\gamma }(x)=\frac{1}{(1+|x|{)}^{\gamma }}$$

$$L_{w_{\gamma }}^2=L^2(w_{\gamma }dx).$$

$$(NS)\{\begin{array}{c}{\partial }_t\mathbf{u}=\Delta \mathbf{u}-(\mathbf{u}\cdot \nabla )\mathbf{u}-\nabla p+\nabla \cdot \mathbb{F}\\ \\ \nabla \cdot \mathbf{u}=0,\mathbf{u}(0,.)={\mathbf{u}}_0\end{array}$$

$$p=\sum _{i=1}^3\sum _{j=1}^3{R}_i{R}_j(u_i{u}_j-F_{i,j})$$

$$\underset{t\to 0}{\lim }\Vert \mathbf{u}(t,.)-{\mathbf{u}}_0{\Vert }_{L_{w_{\gamma }}^2}=0.$$

$${\partial }_t(\frac{|\mathbf{u}{|}^2}{2})=\Delta (\frac{|\mathbf{u}{|}^2}{2})-|\nabla \mathbf{u}{|}^2-\nabla \cdot \left((\frac{|\mathbf{u}{|}^2}{2}+p)\mathbf{u}\right)+\mathbf{u}\cdot (\nabla \cdot \mathbb{F})-\mu .$$

$$\begin{array}{rl}\Vert \mathbf{u}(t,.){\Vert }_{L_{w_{\gamma }}^2}^2& +2{\int }_0^t\Vert \nabla \mathbf{u}(s,.){\Vert }_{L_{w_{\gamma }}^2}^{2}ds\\ \le & \Vert {\mathbf{u}}_0{\Vert }_{L_{w_{\gamma }}^2}^2-{\int }_0^t\int \nabla |\mathbf{u}{|}^2\cdot \nabla w_{\gamma }dxds+{\int }_0^t\int (|\mathbf{u}{|}^2+2p)\mathbf{u}\cdot \nabla (w_{\gamma })dxds\\ & -2\sum _{i=1}^3\sum _{j=1}^3{\int }_0^t\int F_{i,j}({\partial }_i{u}_j)w_{\gamma }+F_{i,j}{u}_i{\partial }_j(w_{\gamma })dxds\end{array}$$

$$\Vert \mathbf{u}(t,.){\Vert }_{L_{w_{\gamma }}^2}^2\le \Vert {\mathbf{u}}_0{\Vert }_{L_{w_{\gamma }}^2}^2+C_{\gamma }{\int }_0^t\Vert \mathbb{F}(s,.){\Vert }_{L_{w_{\gamma }}^2}^{2}ds+C_{\gamma }{\int }_0^t\Vert \mathbf{u}(s,.){\Vert }_{L_{w_{\gamma }}^2}^2+\Vert \mathbf{u}(s,.){\Vert }_{L_{w_{\gamma }}^2}^{6}ds$$

$$(AD)\{\begin{array}{c}{\partial }_t\mathbf{u}=\Delta \mathbf{u}-(\mathbf{b}\cdot \nabla )\mathbf{u}-\nabla p+\nabla \cdot \mathbb{F}\\ \\ \nabla \cdot \mathbf{u}=0,\mathbf{u}(0,.)={\mathbf{u}}_0\end{array}$$

$$p=\sum _{i=1}^3\sum _{j=1}^3{R}_i{R}_j(b_i{u}_j-F_{i,j})$$

$$\underset{t\to 0}{\lim }\Vert \mathbf{u}(t,.)-{\mathbf{u}}_0{\Vert }_{L_{w_{\gamma }}^2}=0.$$

$${\partial }_t(\frac{|\mathbf{u}{|}^2}{2})=\Delta (\frac{|\mathbf{u}{|}^2}{2})-|\nabla \mathbf{u}{|}^2-\nabla \cdot \left(\frac{|\mathbf{u}{|}^2}{2}\mathbf{b}\right)-\nabla \cdot (p\mathbf{u})+\mathbf{u}\cdot (\nabla \cdot \mathbb{F})-\mu .$$

$$\begin{array}{rl}\Vert \mathbf{u}(t,.){\Vert }_{L_{w_{\gamma }}^2}^2& +2{\int }_0^t\Vert \nabla \mathbf{u}(s,.){\Vert }_{L_{w_{\gamma }}^2}^{2}ds\\ \le & \Vert {\mathbf{u}}_0{\Vert }_{L_{w_{\gamma }}^2}^2-{\int }_0^t\int \nabla |\mathbf{u}{|}^2\cdot \nabla w_{\gamma }dxds+{\int }_0^t\int |\mathbf{u}{|}^2\mathbf{b}\cdot \nabla (w_{\gamma })dxds\\ & +2{\int }_0^t\int p\mathbf{u}\cdot \nabla (w_{\gamma })dxds-2\sum _{i=1}^3\sum _{j=1}^3{\int }_0^t\int F_{i,j}({\partial }_i{u}_j)w_{\gamma }+F_{i,j}{u}_i{\partial }_j(w_{\gamma })dxds\end{array}$$

$$\begin{array}{rl}\Vert \mathbf{u}(t,.){\Vert }_{L_{w_{\gamma }}^2}^2& +{\int }_0^t\Vert \nabla \mathbf{u}{\Vert }_{L_{w_{\gamma }}^2}^{2}ds\\ \le & \Vert {\mathbf{u}}_0{\Vert }_{L_{w_{\gamma }}^2}^2+C_{\gamma }{\int }_0^t\Vert \mathbb{F}(s,.){\Vert }_{L_{w_{\gamma }}^2}^{2}ds+C_{\gamma }{\int }_0^t(1+\Vert \mathbf{b}(s,.){\Vert }_{L_{w_{3\gamma /2}}^3}^2)\Vert \mathbf{u}(s,.){\Vert }_{L_{w_{\gamma }}^2}^{2}ds\end{array}$$

$$(A{D}_n)\{\begin{array}{c}{\partial }_t{\mathbf{u}}_n=\Delta {\mathbf{u}}_n-({\mathbf{b}}_n\cdot \nabla ){\mathbf{u}}_n-\nabla p_n+\nabla \cdot {\mathbb{F}}_n\\ \\ \nabla \cdot {\mathbf{u}}_n=0,{\mathbf{u}}_n(0,.)={\mathbf{u}}_{0,n}\end{array}$$

$$p_n=\sum _{i=1}^3\sum _{j=1}^3{R}_i{R}_j(b_{n,i}{u}_{n,j}-F_{n,i,j})$$

$$\underset{t\to 0}{\lim }\Vert {\mathbf{u}}_n(t,.)-{\mathbf{u}}_{0,n}{\Vert }_{L_{w_{\gamma }}^2}=0.$$

$${\partial }_t(\frac{|{\mathbf{u}}_n{|}^2}{2})=\Delta (\frac{|{\mathbf{u}}_n{|}^2}{2})-|\nabla {\mathbf{u}}_n{|}^2-\nabla \cdot \left(\frac{|{\mathbf{u}}_n{|}^2}{2}{\mathbf{b}}_n\right)-\nabla \cdot (p_n{\mathbf{u}}_n)+{\mathbf{u}}_n\cdot (\nabla \cdot {\mathbb{F}}_n)-{\mu }_n.$$

$$\underset{k\to +\infty }{\lim }{\int }_0^{T_0}{\int }_{|y|<R}|{\mathbf{u}}_{n_k}(s,y)-{\mathbf{u}}_{\infty }(s,y){|}^{2}dsdy=0.$$

$$(A{D}_{\infty })\{\begin{array}{c}{\partial }_t{\mathbf{u}}_{\infty }=\Delta {\mathbf{u}}_{\infty }-({\mathbf{b}}_{\infty }\cdot \nabla ){\mathbf{u}}_{\infty }-\nabla p_{\infty }+\nabla \cdot {\mathbb{F}}_{\infty }\\ \\ \nabla \cdot {\mathbf{u}}_{\infty }=0,{\mathbf{u}}_{\infty }(0,.)={\mathbf{u}}_{0,\infty }\end{array}$$

$$\underset{t\to 0}{\lim }\Vert {\mathbf{u}}_{\infty }(t,.)-{\mathbf{u}}_{0,\infty }{\Vert }_{L_{w_{\gamma }}^2}=0.$$

$${\partial }_t(\frac{|{\mathbf{u}}_{\infty }{|}^2}{2})=\Delta (\frac{|{\mathbf{u}}_{\infty }{|}^2}{2})-|\nabla {\mathbf{u}}_{\infty }{|}^2-\nabla \cdot \left(\frac{|{\mathbf{u}}_{\infty }{|}^2}{2}{\mathbf{b}}_{\infty }\right)-\nabla \cdot (p_{\infty }{\mathbf{u}}_{\infty })+{\mathbf{u}}_{\infty }\cdot (\nabla \cdot {\mathbb{F}}_{\infty })-{\mu }_{\infty }.$$

$$|\nabla w_{\delta }(x)|=\delta \frac{w_{\delta }(x)}{1+|x|}$$

$$\underset{x\in {\mathbb{R}}^3,R>0}{\sup }{\left(\frac{1}{|B(x,R)|}{\int }_{B(x,R)}w(y)dy\right)}^{\frac{1}{p}}{\left(\frac{1}{|B(x,R)|}{\int }_{B(x,R)}\frac{dy}{w(y{)}^{\frac{1}{p-1}}}\right)}^{1-\frac{1}{p}}<+\infty .$$

$$\begin{array}{rl}& {\left(\frac{1}{|B(x,R)|}{\int }_{B(x,R)}w(y)dy\right)}^{\frac{1}{p}}{\left(\frac{1}{|B(x,R)|}{\int }_{B(0,R)}\frac{dy}{w(y{)}^{\frac{1}{p-1}}}\right)}^{1-\frac{1}{p}}\\ \le & {\left(\frac{1}{|B(0,R)|}{\int }_{B(x,11R)}w(y)dy\right)}^{\frac{1}{p}}{\left(\frac{1}{|B(0,R)|}{\int }_{B(0,11R)}\frac{dy}{w(y{)}^{\frac{1}{p-1}}}\right)}^{1-\frac{1}{p}}\\ =& {\left(\frac{1}{R^3}{\int }_0^{11R}{r}^2\frac{dr}{(1+r{)}^{\delta }}\right)}^{\frac{1}{p}}{\left(\frac{1}{R^3}{\int }_0^{11R}{r}^2(1+r{)}^{\frac{\delta }{p-1}}dr\right)}^{1-\frac{1}{p}}\\ \le & c_{\delta ,p}{\left(\frac{1}{R^3}{\int }_0^{11R}{r}^2\frac{dr}{r^{\delta }}\right)}^{\frac{1}{p}}\left({\left(\frac{1}{R^3}{\int }_0^{11R}{r}^{2}dr\right)}^{1-\frac{1}{p}}+{\left(\frac{1}{R^3}{\int }_0^{11R}{r}^{2+\frac{\delta }{p-1}}dr\right)}^{1-\frac{1}{p}}\right)\\ =& c_{\delta ,p}\frac{11^3}{(3-\delta {)}^{\frac{1}{p}}}\left(\frac{(11R{)}^{-\frac{\delta }{p}}}{3^{1-\frac{1}{p}}}+\frac{1}{(3+\frac{\delta }{p-1}{)}^{1-\frac{1}{p}}}\right).\end{array}$$

$$\Vert R_{j}f{\Vert }_{L_{w_{\delta }}^p}\le C_{p,\delta }\Vert f{\Vert }_{L_{w_{\delta }}^p}\text{ and }\Vert {\mathcal{M}}_f{\Vert }_{L_{w_{\delta }}^p}\le C_{p,\delta }\Vert f{\Vert }_{L_{w_{\delta }}^p}.$$

$$\Vert \theta \ast f{\Vert }_{L_{w_{\delta }}^p}\le C_{p,\delta }\Vert f{\Vert }_{L_{w_{\delta }}^p}\Vert \theta {\Vert }_1.$$

$$|\theta \ast f(x)|\le \Vert \theta {\Vert }_1{\mathcal{M}}_f(x)$$

$$\{\begin{array}{c}{\partial }_t\mathbf{u}=\Delta \mathbf{u}-(\mathbf{b}\cdot \nabla )\mathbf{u}-\nabla q+\nabla \cdot \mathbb{F}\\ \\ \nabla \cdot \mathbf{u}=0,\end{array}$$

$$\nabla q=\nabla \left(\sum _{i=1}^3\sum _{j=1}^3{R}_i{R}_j(b_i{u}_j-F_{i,j})\right)$$

Full text

**Keywords : ** Navier–Stokes equations, weighted spaces, discretely self-similar solutions, energy controls

**AMS classification : ** 35Q30, 76D05.

1 Introduction.

Infinite-energy weak Leray solutions to the Navier–Stokes equations were introduced by Lemarié-Rieusset in 1999 [8] (they are presented more completely in [9] and [10]). This has allowed to show the existence of local weak solutions for a uniformly locally square integrable initial data.

Other constructions of infinite-energy solutions for locally uniformly square integrable initial data were given in 2006 by Basson [1] and in 2007 by Kikuchi and Seregin [7]. These solutions allowed Jia and Sverak [6] to construct in 2014 the self-similar solutions for large (homogeneous of degree -1) smooth data. Their result has been extended in 2016 by Lemarié-Rieusset [10] to solutions for rough locally square integrable data. We remark that an homogeneous (of degree -1) and locally square integrable data is automatically uniformly locally $L^{2}$.

Recently, Bradshaw and Tsai [2] and Chae and Wolf [3] considered the case of solutions which are self-similar according to a discrete subgroup of dilations. Those solutions are related to an initial data which is self-similar only for a discrete group of dilations; in contrast to the case of self-similar solutions for all dilations, such an initial data, when locally $L^{2}$, is not necessarily uniformly locally $L^{2}$, therefore their results are no consequence of constructions described by Lemarié-Rieusset in [10].

In this paper, we construct an alternative theory to obtain infinite-energy global weak solutions for large initial data, which include the discretely self-similar locally square integrable data. More specifically, we consider the weights

[TABLE]

with $0<\gamma$, and the spaces

[TABLE]

Our main theorem is the following one :

Theorem 1

Let $0<\gamma\leq 2$. If ${\bf u}_{0}$ is a divergence-free vector field such that ${\bf u}_{0}\in L^{2}_{w_{\gamma}}(\mathbb{R}^{3})$ and if $\mathbb{F}$ is a tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that $\mathbb{F}\in L^{2}((0,+\infty),L^{2}_{w_{\gamma}})$, then the Navier–Stokes equations with initial value ${\bf u}_{0}$

[TABLE]

has a global weak solution ${\bf u}$ such that :

• $\bullet$

for every $0<T<+\infty$, ${\bf u}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

the pressure $p$ is related to ${\bf u}$ and $\mathbb{F}$ through the Riesz transforms $R_{i}=\frac{\partial_{i}}{\sqrt{-\Delta}}$ by the formula

[TABLE]

where, for every $0<T<+\infty$, $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}(u_{i}u_{j})$ belongs to $L^{4}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ and $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}F_{i,j}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

the map $t\in[0,+\infty)\mapsto{\bf u}(t,.)$ is weakly continuous from $[0,+\infty)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

the solution ${\bf u}$ is suitable : there exists a non-negative locally finite measure $\mu$ on $(0,+\infty)\times\mathbb{R}^{3}$ such that

[TABLE]

In particular, we have the energy controls

[TABLE]

and

[TABLE]

A key tool for proving Theorem 1 and for applying it to the study of discretely self-similar solutions is given by the following a priori estimates for an advection-diffusion problem :

Theorem 2

Let $0<\gamma\leq 2$. Let $0<T<+\infty$. Let ${\bf u}_{0}$ be a divergence-free vector field such that ${\bf u}_{0}\in L^{2}_{w_{\gamma}}(\mathbb{R}^{3})$ and $\mathbb{F}$ be a tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that $\mathbb{F}\in L^{2}((0,T),L^{2}_{w_{\gamma}})$. Let ${\bf b}$ be a time-dependent divergence free vector-field (${\bf\nabla}\cdot{\bf b}=0$) such that ${\bf b}\in L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$.

Let ${\bf u}$ be a solution of the following advection-diffusion problem

[TABLE]

be such that :

• $\bullet$

${\bf u}$* belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$*

• $\bullet$

the pressure $p$ is related to ${\bf u}$, ${\bf b}$ and $\mathbb{F}$ through the Riesz transforms $R_{i}=\frac{\partial_{i}}{\sqrt{-\Delta}}$ by the formula

[TABLE]

where $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}(b_{i}u_{j})$ belongs to $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ and $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}F_{i,j}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

the map $t\in[0,T)\mapsto{\bf u}(t,.)$ is weakly continuous from $[0,T)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

there exists a non-negative locally finite measure $\mu$ on $(0,T)\times\mathbb{R}^{3}$ such that

[TABLE]

Then, we have the energy controls

[TABLE]

and

[TABLE]

where $C_{\gamma}$ depends only on $\gamma$ (and not on $T$, and not on ${\bf b}$, ${\bf u}$, ${\bf u}_{0}$ nor $\mathbb{F}$).

In particular, we shall prove the following stability result :

Theorem 3

Let $0<\gamma\leq 2$. Let $0<T<+\infty$. Let ${\bf u}_{0,n}$ be divergence-free vector fields such that ${\bf u}_{0,n}\in L^{2}_{w_{\gamma}}(\mathbb{R}^{3})$ and $\mathbb{F}_{n}$ be tensors such that $\mathbb{F}_{n}\in L^{2}((0,T),L^{2}_{w_{\gamma}})$. Let ${\bf b}_{n}$ be time-dependent divergence free vector-fields such that ${\bf b}_{n}\in L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$.

Let ${\bf u}_{n}$ be solutions of the following advection-diffusion problems

[TABLE]

such that :

• $\bullet$

${\bf u}_{n}$* belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{n}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$*

• $\bullet$

the pressure $p_{n}$ is related to ${\bf u}_{n}$, ${\bf b}_{n}$ and $\mathbb{F}_{n}$ by the formula

[TABLE]

• $\bullet$

the map $t\in[0,T)\mapsto{\bf u}_{n}(t,.)$ is weakly continuous from $[0,T)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

there exists a non-negative locally finite measure $\mu_{n}$ on $(0,T)\times\mathbb{R}^{3}$ such that

[TABLE]

If ${\bf u}_{0,n}$ is strongly convergent to ${\bf u}_{0,\infty}$ in $L^{2}_{w_{\gamma}}$, if the sequence $\mathbb{F}_{n}$ is strongly convergent to $\mathbb{F}_{\infty}$ in $L^{2}((0,T),L^{2}_{w_{\gamma}})$, and if the sequence ${\bf b}_{n}$ is bounded in $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$, then there exists $p_{\infty}$, ${\bf u}_{\infty}$, ${\bf b}_{\infty}$ and an increasing sequence $(n_{k})_{k\in\mathbb{N}}$ with values in $\mathbb{N}$ such that

• $\bullet$

${\bf u}_{n_{k}}$* converges -weakly to ${\bf u}_{\infty}$ in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$, ${\bf\nabla}{\bf u}_{n_{k}}$ converges weakly to ${\bf\nabla}{\bf u}_{\infty}$ in $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

${\bf b}_{n_{k}}$* converges weakly to ${\bf b}_{\infty}$ in $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$, $p_{n_{k}}$ converges weakly to $p_{\infty}$ in $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})+L^{2}((0,T),L^{2}_{w_{\gamma}})$*

• $\bullet$

${\bf u}_{n_{k}}$* converges strongly to ${\bf u}_{\infty}$ in $L^{2}_{\rm loc}([0,T)\times\mathbb{R}^{3})$ : for every $T_{0}\in(0,T)$ and every $R>0$, we have*

[TABLE]

Moreover, ${\bf u}_{\infty}$ is a solution of the advection-diffusion problem

[TABLE]

and is such that :

• $\bullet$

the map $t\in[0,T)\mapsto{\bf u}_{\infty}(t,.)$ is weakly continuous from $[0,T)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

there exists a non-negative locally finite measure $\mu_{\infty}$ on $(0,T)\times\mathbb{R}^{3}$ such that

[TABLE]

Notations.

All along the text, $C_{\gamma}$ is a positive constant whose value may change from line to line but which depends only on $\gamma$.

2 The weights $w_{\delta}$.

We consider the weights $w_{\delta}=\frac{1}{(1+|x|)^{\delta}}$ where $0<\delta$ and $x\in\mathbb{R}^{3}$. A very important feature of those weights is the control of their gradients :

[TABLE]

Lemma 1** (Muckenhoupt weights)**

If $0<\delta<3$ and $1<p<+\infty$, then $w_{\delta}$ belongs to the Muckenhoupt class $\mathcal{A}_{p}$.

**Proof : **We recall that a weight $w$ belongs to $\mathcal{A}_{p}(\mathbb{R}^{3})$ for $1<p<+\infty$ if and only if it satisfies the reverse Hölder inequality

[TABLE]

For all $0<R\leq 1$ the inequality $|x-y|<R$ implies $\frac{1}{2}(1+|x|)\leq 1+|y|\leq 2(1+|x|)$, thus we can control the left side in (3) for $w_{\delta}$ by $4^{\frac{\delta}{p}}$.

For all $R>1$ and $|x|>10R$, we have that the inequality $|x-y|<R$ implies $\frac{9}{10}(1+|x|)\leq 1+|y|\leq\frac{11}{10}(1+|x|)$, thus we can control the left side in (3) for $w_{\delta}$ by $(\frac{11}{9})^{\frac{\delta}{p}}$.

Finally, for $R>1$ and $|x|\leq 10R$, we write

[TABLE]

The lemma is proved. $\diamond$

Lemma 2

If $0<\delta<3$ and $1<p<+\infty$, then the Riesz transforms $R_{i}$ and the Hardy–Littlewood maximal function operator are bounded on $L^{p}_{w_{\delta}}=L^{p}(w_{\delta}(x)\,dx)$ :

[TABLE]

**Proof : ** The boundedness of the Riesz transforms or of the Hardy–Littlewwod maximal function on $L^{p}(w_{\gamma}\,dx)$ are basic properties of the Muckenhoupt class $\mathcal{A}_{p}$ [5]. $\diamond$

We will use strategically the next corollary, which is specially useful to obtain discretely self-similar solutions.

Corollary 1** (Non-increasing kernels)**

Let $\theta\in L^{1}(\mathbb{R}^{3})$ be a non-negative radial function which is radially non-increasing. Then, if $0<\delta<3$ and $1<p<+\infty$, we have, for $f\in L^{p}_{w_{\delta}}$, the inequality

[TABLE]

**Proof : **We have the well-known inequality for radial non-increasing kernels [4]

[TABLE]

so that we may conclude with Lemma 2. $\diamond$

We illustrate the utility of Lemma 2 with the following corollaries:

Corollary 2

Let $0<\gamma<\frac{5}{2}$ and $0<T<+\infty$. Let $\mathbb{F}$ be a tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that $\mathbb{F}\in L^{2}((0,T),L^{2}_{w_{\gamma}})$. Let ${\bf b}$ be a time-dependent divergence free vector-field (${\bf\nabla}\cdot{\bf b}=0$) such that ${\bf b}\in L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$.

Let ${\bf u}$ be a solution of the following advection-diffusion problem

[TABLE]

*be such that : ${\bf u}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$, and the pressure $q$ belongs to $\mathcal{D}^{\prime}((0,T)\times\mathbb{R}^{3})$.

Then, the gradient of the pressure ${\bf\nabla}q$ is necessarily related to ${\bf u}$, ${\bf b}$ and $\mathbb{F}$ through the Riesz transforms $R_{i}=\frac{\partial_{i}}{\sqrt{-\Delta}}$ by the formula

[TABLE]

and $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}(b_{i}u_{j})$ belongs to $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ and $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}F_{i,j}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$.

**Proof : ** We define

[TABLE]

As $0<\gamma<\frac{5}{2}$ we can use Lemma 2 and (2) to obtain $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}(b_{i}u_{j})$ belongs to $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ and $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}F_{i,j}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$.

Taking the divergence in (4), we obtain $\Delta(q-p)=0$. We take a test function $\alpha\in\mathcal{D}(\mathbb{R})$ such that $\alpha(t)=0$ for all $|t|\geq\varepsilon$, and a test function $\beta\in\mathcal{D}(\mathbb{R}^{3})$; then the distribution ${\bf\nabla}q*(\alpha\otimes\beta)$ is well defined on $(\varepsilon,T-\varepsilon)\times\mathbb{R}^{3}$.

We fix $t\in(\varepsilon,T-\varepsilon)$ and define

[TABLE]

We have

[TABLE]

Convolution with a function in $\mathcal{D}(\mathbb{R}^{3})$ is a bounded operator on $L^{2}_{w_{\gamma}}$ and on $L^{6/5}_{w_{6\gamma/5}}$ (as, for $\varphi\in\mathcal{D}(\mathbb{R}^{3})$ we have $|f*\varphi|\leq C_{\varphi}\mathcal{M}_{f}$). Thus, we may conclude from (5) that $A_{\alpha,\beta,t}\in L^{2}_{w_{\gamma}}+L^{6/5}_{w_{6\gamma/5}}$. If $\max\{\gamma,\frac{\gamma+2}{2}\}<\delta<5/2$ , we have $A_{\alpha,\beta,t}\in L^{6/5}_{w_{6\delta/5}}$.

In particular, $A_{\alpha,\beta,t}$ is a tempered distribution. As we have

[TABLE]

we find that $A_{\alpha,\beta,t}$ is a polynomial. We remark that for all $1<r<+\infty$ and $0<\delta<3$, $L^{r}_{w_{\delta}}$ does not contain non-trivial polynomials. Thus, $A_{\alpha,\beta,t}=0$. We then use an approximation of identity $\frac{1}{\epsilon^{4}}\alpha(\frac{t}{\epsilon})\beta(\frac{x}{\epsilon})$ and conclude that ${\bf\nabla}(q-p)=0$. $\diamond$

Actually, we can answer a question posed by Bradshaw and Tsai in [2] about the nature of the pressure for self-similar solutions of the Navier–Stokes equations. In effect, we have the next corollary:

Corollary 3

Let $1<\gamma<\frac{5}{2}$ and $0<T<+\infty$. Let $\mathbb{F}$ be a tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that $\mathbb{F}\in L^{2}((0,T),L^{2}_{w_{\gamma}})$.

Let ${\bf u}$ be a solution of the following problem

[TABLE]

*be such that : ${\bf u}$ belongs to $L^{\infty}([0,+\infty),L^{2})_{loc}$ and ${\bf\nabla}{\bf u}$ belongs to $L^{2}([0,+\infty),L^{2})_{loc}$, and the pressure $q$ is in $\mathcal{D}^{\prime}((0,T)\times\mathbb{R}^{3})$.

We suppose that there exists $\lambda>1$ such that $\lambda^{2}\mathbb{F}(\lambda^{2}t,\lambda x)=\mathbb{F}(t,x)$ and $\lambda{\bf u}(\lambda^{2}t,\lambda x)={\bf u}(t,x)$. Then, the gradient of the pressure ${\bf\nabla}q$ is necessarily related to ${\bf u}$ and $\mathbb{F}$ through the Riesz transforms $R_{i}=\frac{\partial_{i}}{\sqrt{-\Delta}}$ by the formula

[TABLE]

and $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}(u_{i}u_{j})$ belongs to $L^{4}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ and $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}F_{i,j}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$.

**Proof : ** We shall use Corollary 2, and thus we need to show that ${\bf u}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}}\cap L^{3}((0,T),L^{3}_{3\gamma/2}))$ and ${\bf\nabla}{\bf u}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$. In fact,

[TABLE]

and

[TABLE]

For ${\bf\nabla}{\bf u}$, we compute for $k\in\mathbb{N}$,

[TABLE]

We may conclude that ${\bf\nabla}{\bf u}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$, since for $\gamma>1$ we have $\sum_{k\in\mathbb{N}}\lambda^{(1-\gamma)k}<+\infty$.

Now, we use the Sobolev embeddings described in next Lemma (Lemma 3) to get that ${\bf u}$ belongs to $L^{2}((0,T),L^{6}_{w_{3\gamma}})$, and thus (by interpolation with $L^{\infty}((0,T),L^{2}_{w_{\gamma}}))$ to $L^{4}((0,T),L^{3}_{w_{3\gamma/2}})$.

In particular, $\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}(u_{i}u_{j})$ belongs to $L^{4}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$, since we have

[TABLE]

$\diamond$

Lemma 3** (Sobolev embeddings)**

Let $\delta>0$. If $f\in L^{2}_{w_{\delta}}$ and ${\bf\nabla}f\in L^{2}_{w_{\delta}}$ then $f\in L^{6}_{w_{3\delta}}$ and

[TABLE]

**Proof : **Since both $f$ and $w_{\delta/2}$ are locally in $H^{1}$, we write

[TABLE]

and thus

[TABLE]

Thus, $w_{\delta/2}f$ belongs to $L^{6}$ (since $H^{1}\subset L^{6}$), or equivalently $f\in L^{6}_{w_{3\delta}}$. $\diamond$

3 A priori estimates for the advection-diffusion problem.

3.1 Proof of Theorem 2.

Let $0<t_{0}<t_{1}<T$. We take a function $\alpha\in\mathcal{C}^{\infty}(\mathbb{R})$ which is non-decreasing, with $\alpha(t)$ equal to [math] for $t<1/2$ and equal to $1$ for $t>1$. For $0<\eta<\min(\frac{t_{0}}{2},T-t_{1})$, we define

[TABLE]

We take as well a non-negative function $\phi\in\mathcal{D}(\mathbb{R}^{3})$ which is equal to $1$ for $|x|\leq 1$ and to [math] for $|x|\geq 2$. For $R>0$, we define $\phi_{R}(x)=\phi(\frac{x}{R})$. Finally, we define, for $\epsilon>0$, $w_{\gamma,\epsilon}=\frac{1}{(1+\sqrt{\epsilon^{2}+|x|^{2}})^{\delta}}$. We have $\alpha_{\eta,t_{0},t_{1}}(t)\phi_{R}(x)w_{\gamma,\epsilon}(x)\in\mathcal{D}((0,T)\times\mathbb{R}^{3})$ and $\alpha_{\eta,t_{0},t_{1}}(t)\phi_{R}(x)w_{\gamma,\epsilon}(x)\geq 0$. Thus, using the local energy balance (1) and the fact that $\mu\geq 0$, we find

[TABLE]

We remark that, independently from $R>1$ and $\epsilon>0$, we have (for $0<\gamma\leq 2$)

[TABLE]

Moreover, we know that ${\bf u}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})\cap L^{2}((0,T),L^{6}_{w_{3\gamma}})$ hence to $L^{4}((0,T),L^{3}_{w_{3\gamma/2}})$. Since $T<+\infty$, we have as well ${\bf u}\in L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$. (This is the same type of integrability as required for ${\bf b}$). Moreover, we have $pu_{i}\in L^{1}_{w_{3\gamma/2}}$ since $w_{\gamma}p\in L^{2}((0,T),L^{6/5}+L^{2})$ and $w_{\gamma/2}{\bf u}\in L^{2}((0,T),L^{2}\cap L^{6})$. All those remarks will allow us to use dominated convergence.

We first let $\eta$ go to [math]. We find that

[TABLE]

Let us define

[TABLE]

As we have

[TABLE]

we find that, when $t_{0}$ and $t_{1}$ are Lebesgue points of the measurable function $A_{R,\epsilon}$

[TABLE]

Then, by continuity, we can let $t_{0}$ go to [math] and thus replace $t_{0}$ by [math] in the inequality. Moreover, if we let $t_{1}$ go to $t$, then by weak continuity, we find that $A_{R,\epsilon}(t)\leq\lim_{t_{1}\rightarrow t}A_{R,\epsilon}(t_{1})$, so that we may as well replace $t_{1}$ by $t\in(0,T)$. Thus we find that for every $t\in(0,T)$, we have

[TABLE]

Thus, letting $R$ go to $+\infty$ and then $\epsilon$ go to [math], we find by dominated convergence that, for every $t\in(0,T)$, we have

[TABLE]

Now we write

[TABLE]

Writing

[TABLE]

and using the fact that $w_{6\gamma/5}\in\mathcal{A}_{6/5}$ and $w_{\gamma}\in\mathcal{A}_{2}$, we get

[TABLE]

and

[TABLE]

Finally, we have

[TABLE]

We have obtained

[TABLE]

and Theorem 2 is proven. $\diamond$

3.2 Passive transportation.

From inequality (7), we have the following direct consequence :

Corollary 4

Under the assumptions of Theorem 2, we have

[TABLE]

and

[TABLE]

where the constant $C_{\gamma}$ depends only on $\gamma$.

Another direct consequence is the following uniqueness result for the advection-diffusion problem with a (locally in time), bounded ${\bf b}$ :

Corollary 5

. Let $0<\gamma\leq 2$. Let $0<T<+\infty$. Let ${\bf u}_{0}$ be a divergence-free vector field such that ${\bf u}_{0}\in L^{2}_{w_{\gamma}}(\mathbb{R}^{3})$ and $\mathbb{F}$ be a tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that $\mathbb{F}\in L^{2}((0,T),L^{2}_{w_{\gamma}})$. Let ${\bf b}$ be a time-dependent divergence free vector-field (${\bf\nabla}\cdot{\bf b}=0$) such that ${\bf b}\in L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$. Assume moreover that ${\bf b}$ belongs to $L^{2}_{t}L^{\infty}_{x}(K)$ for every compact subset $K$ of $(0,T)\times\mathbb{R}^{3}$.

Let $({\bf u}_{1},p_{1})$ and $({\bf u}_{2},p_{2})$ be two solutions of the following advection-diffusion problem

[TABLE]

be such that, for $k=1$ and $k=2$, :

• $\bullet$

${\bf u}_{k}$* belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{k}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$*

• $\bullet$

the pressure $p_{k}$ is related to ${\bf u}_{k}$, ${\bf b}$ and $\mathbb{F}$ through the Riesz transforms $R_{i}=\frac{\partial_{i}}{\sqrt{-\Delta}}$ by the formula

[TABLE]

• $\bullet$

the map $t\in[0,T)\mapsto{\bf u}_{k}(t,.)$ is weakly continuous from $[0,T)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

Then ${\bf u}_{1}={\bf u}_{2}$.

**Proof : **Let ${\bf v}={\bf u}_{1}-{\bf u}_{2}$ and $q=p_{1}-p_{2}$. Then we have

[TABLE]

Moreover on every compact subset $K$ of $(0,T)\times\mathbb{R}^{3}$, ${\bf b}\otimes{\bf v}$ is in $L^{2}_{t}L^{2}_{x}$, while it belongs globally to $L^{3}_{t}L^{6/5}_{w_{6\gamma/5}}$. Writing, for $\varphi,\psi\in\mathcal{D}((0,T)\times\mathbb{R}^{3})$ such that $\psi=1$ on the neigborhood of the support of $\varphi$,

[TABLE]

we find that $\|q_{1}\|_{L^{2}L^{2}}\leq C_{\varphi,\psi}\|\psi{\bf b}\otimes{\bf v}\|_{L^{2}L^{2}}$ and

[TABLE]

with

[TABLE]

Thus, we may take the scalar product of $\partial_{t}{\bf v}$ with ${\bf v}$ and find that

[TABLE]

Thus we are under the assumptions of Theorem 2 and we may use Corollary 4 to find that ${\bf v}=0$. $\diamond$

3.3 Active transportation.

We begin with the following lemma :

Lemma 4

Let $\alpha$ be a non-negative bounded measurable function on $[0,T)$ such that, for two constants $A,B\geq 0$, we have

[TABLE]

If $T_{0}>0$ and $T_{1}=\min(T,T_{0},\frac{1}{4B(A+BT_{0})^{2}})$, we have, for every $t\in[0,T_{1}]$, $\alpha(t)\leq\sqrt{2}(A+BT_{0})$.

**Proof : **We write $\alpha\leq 1+\alpha^{3}$. We define

[TABLE]

We have, for $t\in[0,T_{1}]$, $\alpha\leq\Phi\leq\Psi$. Since $\Psi$ is $\mathcal{C}^{1}$, we may write

[TABLE]

and thus

[TABLE]

We thus find

[TABLE]

The lemma is proven. $\diamond$

Corollary 6

Assume that ${\bf u}_{0}$, ${\bf u}$, $p$, $\mathbb{F}$ and ${\bf b}$ satisfy assumptions of Theorem 2, Assume moreover that ${\bf b}$ is controlled by ${\bf u}$ : for every $t\in(0,T)$,

[TABLE]

Then there exists a constant $C_{\gamma}\geq 1$ such that if $T_{0}<T$ is such that

[TABLE]

then

[TABLE]

and

[TABLE]

**Proof : **We start from inequality (7) :

[TABLE]

We write

[TABLE]

This gives

[TABLE]

For $t\leq T_{0}$, we get

[TABLE]

and we may conclude with Lemma 4. $\diamond$

4 Stability of solutions for the advection-diffusion problem.

4.1 The Rellich lemma.

We recall the Rellich lemma :

Lemma 5** (Rellich)**

If $s>0$ and $(f_{n})$ is a sequence of functions on $\mathbb{R}^{d}$ such that

• $\bullet$

the family $(f_{n})$ is bounded in $H^{s}(\mathbb{R}^{d})$

• $\bullet$

there is a compact subset of $\mathbb{R}^{d}$ such that the support of each $f_{n}$ is included in $K$

then there exists a subsequence $(f_{n_{k}})$ such that $f_{n_{k}}$ is strongly convergent in $L^{2}(\mathbb{R}^{d})$.

We shall use a variant of this lemma (see [9]) :

Lemma 6** (space-time Rellich)**

If $s>0$, $\sigma\in\mathbb{R}$ and $(f_{n})$ is a sequence of functions on $(0,T)\times\mathbb{R}^{d}$ such that, for all $T_{0}\in(0,T)$ and all $\varphi\in\mathcal{D}(\mathbb{R}^{3})$

• $\bullet$

$\varphi f_{n}$* is bounded in $L^{2}((0,T_{0}),H^{s})$*

• $\bullet$

$\varphi\partial_{t}f_{n}$* is bounded in $L^{2}((0,T_{0}),H^{\sigma})$*

then there exists a subsequence $(f_{n_{k}})$ such that $f_{n_{k}}$ is strongly convergent in $L^{2}_{\rm loc}([0,T)\times\mathbb{R}^{3})$ : if $f_{\infty}$ is the limit, we have for all $T_{0}\in(0,T)$ and all $R_{0}>0$

[TABLE]

**Proof : **With no loss of generality, we may assume that $\sigma<\min(1,s)$. Define $g$ by $g_{n}(t,x)=\alpha(t)\varphi(x)f_{n}(t,x)$ if $t>0$ and $g_{n}(t,x)=\alpha(t)\varphi(x)f_{n}(-t,x)$ if $t<0$, where $\alpha\in\mathcal{C}^{\infty}$ on $(0,T)$, is equal to $1$ on $[0,T_{0}]$ and equal to [math] for $t>\frac{T+T_{0}}{2}$, and $\varphi(x)=1$ on $B(0,R_{0})$. Then the support of $g_{n}$ is contained in $[-\frac{T+T_{0}}{2},\frac{T+T_{0}}{2}]\times{\rm Supp}\,\varphi$. Moreover, $g_{n}$ is bounded in $L^{2}_{t}H^{s}$ and $\partial_{t}g_{n}$ is bounded in $L^{2}H^{\sigma}$ so that $g_{n}$ is bounded in $H^{\rho}(\mathbb{R}\times\mathbb{R}^{3})$ with $\rho=\frac{s}{s+1-\sigma}$ (just write $(1+\tau^{2}+\xi^{2})^{\frac{s}{s+1-\sigma}}\leq\left((1+\tau^{2})(1+\xi^{2})^{\sigma}\right)^{\frac{s}{s+1-\sigma}}\left((1+\xi^{2})^{s}\right)^{\frac{1-\sigma}{s+1-\sigma}}$).. By the Rellich lemma, we know that there is a subsequence $g_{n_{k}}$ which is strongly convergent in $L^{2}(\mathbb{R}\times\mathbb{R}^{3})$, thus a subsequence $f_{n_{k}}$ which is strongly convergent in $L^{2}((0,T_{0})\times B(0,R_{0}))$.

We then iterate this argument for an increasing sequence of times $T_{0}<T_{1}<\dots<T_{N}\rightarrow T$ and an increasing sequence of radii $R_{0}<R_{1}<\dots<R_{N}\rightarrow+\infty$ and finish the proof. by the classical diagonal process of Cantor. $\diamond$

4.2 Proof of Theorem 3.

Assume that ${\bf u}_{0,n}$ is strongly convergent to ${\bf u}_{0,\infty}$ in $L^{2}_{w_{\gamma}}$ and that the sequence $\mathbb{F}_{n}$ is strongly convergent to $\mathbb{F}_{\infty}$ in $L^{2}((0,T),L^{2}_{w_{\gamma}})$, and assume that the sequence ${\bf b}_{n}$ is bounded in $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$. Then, by Theorem 2 and Corollary 4, we know that ${\bf u}_{n}$ is bounded in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{n}$ is bounded in $L^{2}((0,T),L^{2}_{w_{\gamma}})$. In particular, writing $p_{n}=p_{n,1}+p_{n,2}$ with

[TABLE]

we get that $p_{n,1}$ is bounded in $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ and $p_{n,2}$ is bounded in $L^{2}((0,T),L^{2}_{w_{\gamma}})$.

If $\varphi\in\mathcal{D}(\mathbb{R}^{3})$, we find that $\varphi{\bf u}_{n}$ is bounded in $L^{2}((0,T),H^{1})$ and, writing

[TABLE]

$\varphi\partial_{t}{\bf u}_{n}$ is bounded in $L^{2}L^{2}+L^{2}W^{-1,6/5}+L^{2}H^{-1}\subset L^{2}((0,T),H^{-2})$. Thus, by Lemma 6, there exists ${\bf u}_{\infty}$ and an increasing sequence $(n_{k})_{k\in\mathbb{N}}$ with values in $\mathbb{N}$ such that ${\bf u}_{n_{k}}$ converges strongly to ${\bf u}_{\infty}$ in $L^{2}_{\rm loc}([0,T)\times\mathbb{R}^{3})$ : for every $T_{0}\in(0,T)$ and every $R>0$, we have

[TABLE]

As ${\bf u}_{n}$ is bounded in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{n}$ is bounded in $L^{2}((0,T),L^{2}_{w_{\gamma}})$, the convergence of ${\bf u}_{n_{k}}$ to ${\bf u}_{\infty}$ in $\mathcal{D}^{\prime}((0,T)\times\mathbb{R}^{3})$ implies that ${\bf u}_{n_{k}}$ converges *-weakly to ${\bf u}_{\infty}$ in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{n_{k}}$ converges weakly to ${\bf\nabla}{\bf u}_{\infty}$ in $L^{2}((0,T),L^{2}_{w_{\gamma}})$.

By Banach–Alaoglu’s theorem, we may assume that there exists ${\bf b}_{\infty}$ such that ${\bf b}_{n_{k}}$ converges weakly to ${\bf b}_{\infty}$ in $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$. In particular $b_{n_{k},i}u_{n_{k},j}$ is weakly convergent in $(L^{6/5}L^{6/5})_{\rm loc}$ and thus in $\mathcal{D}^{\prime}((0,T)\times\mathbb{R}^{3})$; as it is bounded in $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$, it is weakly convergent in $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ to $b_{\infty,i}u_{\infty,j}$. Let

[TABLE]

As the Riesz transforms are bounded on $L^{6/5}_{w_{\frac{6\gamma}{5}}}$ and on $L^{2}_{w_{\gamma}}$, we find that $p_{n_{k},1}$ is weakly convergent in $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ to $p_{\infty,1}$ and that $p_{n_{k},2}$ is strongly convergent in $L^{2}((0,T),L^{2}_{w_{\gamma}})$ to $p_{\infty,2}$.

In particular, we find that in $\mathcal{D}^{\prime}((0,T)\times\mathbb{R}^{3})$

[TABLE]

In particular, $\partial_{t}{\bf u}_{\infty}$ is locally in $L^{2}H^{-2}$, and thus ${\bf u}_{\infty}$ has representative such that $t\mapsto{\bf u}_{\infty}(t,.)$ is continuous from $[0,T)$ to $\mathcal{D}^{\prime}(\mathbb{R}^{3})$ and coincides with ${\bf u}_{\infty}(0,.)+\int_{0}^{t}\partial_{t}{\bf u}_{\infty}\,ds$. In $\mathcal{D}^{\prime}((0,T)\times\mathbb{R}^{3})$, we have that

[TABLE]

Thus, ${\bf u}_{\infty}(0,.)={\bf u}_{0,\infty}$, and ${\bf u}_{\infty}$ is a solution of $(AD_{\infty})$.

Next, we define

[TABLE]

As ${\bf u}_{n}$ is bounded in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{n}$ is bounded in $L^{2}((0,T),L^{2}_{w_{\gamma}})$, it is bounded in $L^{2}((0,T),L^{6}_{w_{3\gamma/2}})$ and by interpolation with $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ it is bounded in $L^{10/3}((0,T),L^{10/3}_{w_{5\gamma/3}})$. Thus, $u_{n_{k}}$ is locally bounded in $L^{10/3}L^{10/3}$ and locally strongly convergent in $L^{2}L^{2}$; it is then strongly convergent in $L^{3}L^{3}$. Thus, $A_{n_{k}}$ is convergent in $\mathcal{D}^{\prime}((0,T)\times\mathbb{R}^{3})$ to

[TABLE]

In particular, $A_{\infty}=\lim_{n_{k}\rightarrow+\infty}|{\bf\nabla}{\bf u}_{n_{k}}|^{2}+\mu_{n_{k}}$. If $\Phi\in\mathcal{D}((0,T)\times\mathbb{R}^{3})$ is non-negative, we have

[TABLE]

(since $\sqrt{\Phi}{\bf\nabla}{\bf u}_{n_{k}}$ is weakly convergent to $\sqrt{\Phi}{\bf\nabla}{\bf u}_{\infty}$ in $L^{2}L^{2}$). Thus, there exists a non-negative locally finite measure $\mu_{\infty}$ on $(0,T)\times\mathbb{R}^{3}$ such that $A_{\infty}=|{\bf\nabla}{\bf u}_{\infty}|^{2}+\mu_{\infty}$, i.e. such that

[TABLE]

Finally, we start from inequality (6) :

[TABLE]

This gives

[TABLE]

As we have

[TABLE]

we see that ${\bf u}_{n_{k}}(t,.)$ is convergent to ${\bf u}_{\infty}(t,.)$ in $\mathcal{D}^{\prime}(\mathbb{R}^{3})$, hence is weakly convergent in $L^{2}_{\rm loc}$ (as it is bounded in $L^{2}_{w_{\gamma}}$), so that :

[TABLE]

Similarly, as ${\bf\nabla}{\bf u}_{n_{k}}$ is weakly convergent in $L^{2}L^{2}_{w_{\gamma}}$, we have

[TABLE]

Thus, letting $R$ go to $+\infty$ and then $\epsilon$ go to [math], we find by dominated convergence that, for every $t\in(0,T)$, we have

[TABLE]

Letting $t$ go to [math], we find

[TABLE]

On the other hand, we know that ${\bf u}_{\infty}$ is weakly continuous in $L^{2}_{w_{\gamma}}$ and thus we have

[TABLE]

This gives $\|{\bf u}_{0,\infty}\|_{L^{2}_{w_{\gamma}}}^{2}=\lim_{t\rightarrow 0}\|{\bf u}_{\infty}(t,.)\|_{L^{2}_{w_{\gamma}}}^{2}$, which allows to turn the weak convergence into a strong convergence. Theorem 3 is proven. $\diamond$

5 Solutions of the Navier–Stokes problem with initial data in $L^{2}_{w_{\gamma}}$.

We now prove Theorem 1. The idea is to approximate the problem by a Navier–Stokes problem in $L^{2}$, then use the a priori estimates (Theorem 2) and the stability theorem (Theorem 3) to find a solution to the Navier–Stokes problem with data in $L^{2}_{w_{\gamma}})$.

5.1 Approximation by square integrable data.

Lemma 7** (Leray’s projection operator)**

Let $0<\delta<3$ and $1<r<+\infty$. If ${\bf v}$ is a vector field on $\mathbb{R}^{3}$ such that ${\bf v}\in L^{r}_{w_{\delta}}$, then there exists a unique decompostion

[TABLE]

such that

• $\bullet$

${\bf v}_{\sigma}\in L^{r}_{w_{\delta}}$* and ${\bf\nabla}\cdot{\bf v}_{\sigma}=0$.*

• $\bullet$

${\bf v}_{\nabla}\in L^{r}_{w_{\delta}}$* and ${\bf\nabla}\wedge{\bf v}_{\nabla}=0$.*

We shall write ${\bf v}_{\sigma}=\mathbb{P}{\bf v}$, where $\mathbb{P}$ is Leray’s projection operator.

Similarly, if ${\bf v}$ is a distribution vector field of the type ${\bf v}={\bf\nabla}\cdot\mathbb{G}$ with $\mathbb{G}\in L^{r}_{w_{\delta}}$ then there exists a unique decompostion

[TABLE]

such that

• $\bullet$

there exists $\mathbb{H}\in L^{r}_{w_{\delta}}$ such that ${\bf v}_{\sigma}\ ={\bf\nabla}\cdot\mathbb{H}$ and ${\bf\nabla}\cdot{\bf v}_{\sigma}=0$.

• $\bullet$

there exists $q\in L^{r}_{w_{\delta}}$ such that ${\bf v}_{\nabla}={\bf\nabla}q$ (and thus ${\bf\nabla}\wedge{\bf v}_{\nabla}=0$).

We shall still write ${\bf v}_{\sigma}=\mathbb{P}{\bf v}$. Moreover, the function $q$ is given by

[TABLE]

**Proof : **As $w_{\delta}\in\mathcal{A}_{r}$ the Riesz transforms are bounded on $L^{r}_{w_{\delta}}$. Using the identity

[TABLE]

we find (if the decomposition exists) that

[TABLE]

This proves the uniqueness. By linearity, we just have to prove that ${\bf v}=0\implies{\bf v}_{\nabla}=0$. We have $\Delta{\bf v}_{\nabla}=0$, and thus ${\bf v}_{\nabla}$ is harmonic; as it belongs to $\mathcal{S}^{\prime}$, we find that it is a polynomial. But a polynomial which belongs to $L^{r}_{w_{\delta}}$ must be equal to [math]. Similarly, if ${\bf v}_{\nabla}={\bf\nabla}q$, then $\Delta q={\bf\nabla}\cdot{\bf v}_{\nabla}={\bf\nabla}\cdot{\bf v}=0$; thus $q$ is harmonic and belongs to $L^{r}_{w_{\delta}}$, hence $q=0$.

For the existence, it is enough to check that $v_{\nabla,i}=-\sum_{j=1}^{3}R_{i}R_{j}v_{j}$ in the first case and ${\bf v}_{\nabla}={\bf\nabla}q$ with $q=\sum_{i=1}^{3}\sum_{j=1}^{3}R_{i}R_{j}(G_{i,j})$ in the second case fulfill the conclusions of the lemma. $\diamond$

Lemma 8

Let $0<\gamma\leq 2$. Let ${\bf u}_{0}$ be a divergence-free vector field such that ${\bf u}_{0}\in L^{2}_{w_{\gamma}}(\mathbb{R}^{3})$ and $\mathbb{F}$ be a tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that $\mathbb{F}\in L^{2}((0,+\infty),L^{2}_{w_{\gamma}})$. Let $\phi\in\mathcal{D}(\mathbb{R}^{3})$ be a non-negative function which is equal to $1$ for $|x|\leq 1$ and to [math] for $|x|\geq 2$. For $R>0$, we define $\phi_{R}(x)=\phi(\frac{x}{R})$, ${\bf u}_{0,R}=\mathbb{P}(\phi_{R}{\bf u}_{0})$ and $\mathbb{F}_{R}=\phi_{R}\mathbb{F}$. Then ${\bf u}_{0,R}$ is a divergence-free square integrable vector field and $\lim_{R\rightarrow+\infty}\|{\bf u}_{0,R}-{\bf u}_{0}\|_{L^{2}_{w_{\gamma}}}=0$. Similarly, $\mathbb{F}_{R}$ belongs to $L^{2}L^{2}$ and $\lim_{R\rightarrow+\infty}\|\mathbb{F}_{R}-\mathbb{F}\|_{L^{2}((0,+\infty),L^{2}_{w_{\gamma}})}=0$.

**Proof : **By dominated convergence, we have $\lim_{R\rightarrow+\infty}\|\phi_{R}{\bf u}_{0}-{\bf u}_{0}\|_{L^{2}_{w_{\gamma}}}=0$. We conclude by writing ${\bf u}_{0,R}-{\bf u}_{0}=\mathbb{P}(\phi_{R}{\bf u}_{0}-{\bf u}_{0})$. $\diamond$

5.2 Leray’s mollification.

We want to solve the Navier–Stokes equations with initial value ${\bf u}_{0}$ :

[TABLE]

We begin with Leray’s method [11] for solving the problem in $L^{2}$ :

[TABLE]

The idea of Leray is to mollify the non-linearity by replacing ${\bf u}_{R}\cdot{\bf\nabla}$ by $({\bf u}_{R}*\theta_{\epsilon})\cdot{\bf\nabla}$, where $\theta(x)=\frac{1}{\epsilon^{3}}\theta(\frac{x}{\epsilon})$, $\theta\in\mathcal{D}(\mathbb{R}^{3})$, $\theta$ is non-negative and radially decreasing and $\int\theta\,dx=1$. We thus solve the problem

[TABLE]

The classical result of Leray states that the problem $(NS_{R,\epsilon})$ is well-posed :

Lemma 9

Let ${\bf v}_{0}\in L^{2}$ be a divergence-free vector field. Let $\mathbb{G}\in L^{2}((0,+\infty),L^{2})$. Then the problem

[TABLE]

has a unique solution ${\bf v}_{\epsilon}$ in $L^{\infty}((0,+\infty),L^{2})\cap L^{2}((0,+\infty),\dot{H}^{1})$. Moreover, this solution belongs to $\mathcal{C}([0,+\infty),L^{2})$.

5.3 Proof of Theorem 1 (local existence)

We use Lemma 9 and find a solution ${\bf u}_{R,\epsilon}$ to the problem $(NS_{R,\epsilon})$. Then we check that ${\bf u}_{R,\epsilon}$ fulfills the assumptions of Theorem 2 and of Corollary 6 :

• $\bullet$

${\bf u}_{R,\epsilon}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{R,\epsilon}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

the map $t\in[0,+\infty)\mapsto{\bf u}_{R,\epsilon}(t,.)$ is weakly continuous from $[0,+\infty)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

on $(0,T)\times\mathbb{R}^{3}$, ${\bf u}_{R,\epsilon}$ fulfills the energy equality :

[TABLE]

with ${\bf b}_{R,\epsilon}={\bf u}_{R,\epsilon}*\theta_{\epsilon}$.

• $\bullet$

${\bf b}_{R,\epsilon}$ is controlled by ${\bf u}_{R,\epsilon}$ : for every $t\in(0,T)$,

[TABLE]

Thus, we know that, for every time $T_{0}$ such that

[TABLE]

we have

[TABLE]

and

[TABLE]

Moreover, we have that

[TABLE]

so that

[TABLE]

Let $R_{n}\rightarrow+\infty$ and $\epsilon_{n}\rightarrow 0$. Let ${\bf u}_{0,n}={\bf u}_{0,R_{n}}$, $\mathbb{F}_{n}=\mathbb{F}_{R_{n}}$, ${\bf b}_{n}={\bf b}_{R_{n},\epsilon_{n}}$ and ${\bf u}_{n}={\bf u}_{R_{n},\epsilon_{n}}$. We may then apply Theorem 3, since ${\bf u}_{0,n}$ is strongly convergent to ${\bf u}_{0}$ in $L^{2}_{w_{\gamma}}$, $\mathbb{F}_{n}$ is strongly convergent to $\mathbb{F}$ in $L^{2}((0,T_{0}),L^{2}_{w_{\gamma}})$, and the sequence ${\bf b}_{n}$ is bounded in $L^{3}((0,T_{0}),L^{3}_{w_{3\gamma/2}})$. Thus there exists $p$, ${\bf u}$, ${\bf b}$ and an increasing sequence $(n_{k})_{k\in\mathbb{N}}$ with values in $\mathbb{N}$ such that

• $\bullet$

${\bf u}_{n_{k}}$ converges *-weakly to ${\bf u}$ in $L^{\infty}((0,T_{0}),L^{2}_{w_{\gamma}})$, ${\bf\nabla}{\bf u}_{n_{k}}$ converges weakly to ${\bf\nabla}{\bf u}$ in $L^{2}((0,T_{0}),L^{2}_{w_{\gamma}})$

• $\bullet$

${\bf b}_{n_{k}}$ converges weakly to ${\bf b}$ in $L^{3}((0,T_{0}),L^{3}_{w_{3\gamma/2}})$, $p_{n_{k}}$ converges weakly to $p$ in $L^{3}((0,T_{0}),L^{6/5}_{w_{\frac{6\gamma}{5}}})+L^{2}((0,T_{0}),L^{2}_{w_{\gamma}})$

• $\bullet$

${\bf u}_{n_{k}}$ converges strongly to ${\bf u}$ in $L^{2}_{\rm loc}([0,T_{0})\times\mathbb{R}^{3})$.

Moreover, ${\bf u}$ is a solution of the advection-diffusion problem

[TABLE]

and is such that :

• $\bullet$

the map $t\in[0,T_{0})\mapsto{\bf u}(t,.)$ is weakly continuous from $[0,T_{0})$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

there exists a non-negative locally finite measure $\mu$ on $(0,T_{0})\times\mathbb{R}^{3}$ such that

[TABLE]

Finally, as ${\bf b}_{n}=\theta_{\epsilon_{n}}*({\bf u}_{n}-{\bf u})+\theta_{\epsilon_{n}}*{\bf u}$, we see that ${\bf b}_{n_{k}}$ is strongly convergent to ${\bf u}$ in $L^{3}_{\rm loc}([0,T_{0})\times\mathbb{R}^{3})$, so that ${\bf b}={\bf u}$ : thus, ${\bf u}$ is a solution of the Navier–Stokes problem on $(0,T_{0})$. (It is easy to check that

[TABLE]

as $u_{i,n_{k}}u_{j,n_{k}}$ is weakly convergent to $u_{i}u_{j}$ in $L^{4}((0,T_{0}),L^{6/5}_{w_{\frac{6\gamma}{5}}})$ and $w_{\frac{6\gamma}{5}}\in\mathcal{A}_{6/5}$).

5.4 Proof of Theorem 1 (global existence)

In order to finish the proof, we shall use the scaling properties of the Navier–Stokes equations : if $\lambda>0$, then ${\bf u}$ is a solution of the Cauchy initial value problem for the Navier–Stokes equations on $(0,T)$ with initial value ${\bf u}_{0}$ and forcing tensor $\mathbb{F}$ if and only if ${\bf u}_{\lambda}(t,x)=\lambda{\bf u}(\lambda^{2}t,\lambda x)$ is a solution of the Navier–Stokes equations on $(0,T/\lambda^{2})$ with initial value ${\bf u}_{0,\lambda}(x)=\lambda{\bf u}_{0}(\lambda x)$ and forcing tensor $\mathbb{F}_{\lambda}(t,x)=\lambda^{2}\mathbb{F}(\lambda^{2}t,\lambda x)$.

We take $\lambda>1$ and for $n\in\mathbb{N}$ we consider the Navier–Stokes problem with initial value ${\bf v}_{0,n}=\lambda^{n}{\bf u}_{0}(\lambda^{n}\cdot)$ and forcing tensor $\mathbb{F}_{n}=\lambda^{2n}\mathbb{F}(\lambda^{2n}\cdot,\lambda^{n}\cdot)$. Then we have seen that we can find a solution ${\bf v}_{n}$ on $(0,T_{n})$, with

[TABLE]

Of course, we have ${\bf v}_{n}(t,x)=\lambda^{n}{\bf u}_{n}(\lambda^{2n}t,\lambda^{n}x)$ where ${\bf u}_{n}$ is a solution of the Navier–Stokes equations on $(0,\lambda^{2n}T_{n})$ with initial value ${\bf u}_{0}$ and forcing tensor $\mathbb{F}$

Lemma 10

[TABLE]

**Proof : **We have

[TABLE]

We have

[TABLE]

as $\gamma\leq 2$ and we have, by dominated convergence,

[TABLE]

Similarly, we have

[TABLE]

Thus, $\lim_{n\rightarrow+\infty}\lambda^{2n}T_{n}=+\infty$. $\diamond$

Now, for a given $T>0$, if $\lambda^{2n}T_{n}>T$ for $n\geq n_{T}$, then ${\bf u}_{n}$ is a solution of the Navier-Stokes problem on $(0,T)$. Let ${\bf w}_{n}(t,x)=\lambda^{n_{T}}{\bf u}_{n}(\lambda^{2n_{T}}t,\lambda^{n_{T}}x)$. For $n\geq n_{T}$, ${\bf w}_{n}$ is a solution of the Navier-Stokes problem on $(0,\lambda^{-2n_{T}}T)$ with initial value ${\bf v}_{0,n_{T}}$ and forcing tensor $\mathbb{F}_{n_{T}}$. As $\lambda^{-2n_{T}}T\leq T_{n_{T}}$, we have

[TABLE]

By corollary 6, we have

[TABLE]

and

[TABLE]

We have

[TABLE]

and

[TABLE]

Thus, we have a uniform control of ${\bf u}_{n}$ and of ${\bf\nabla}{\bf u}_{n}$ on $(0,T)$ for $n\geq n_{T}$. We may then apply the Rellich lemma (Lemma 6) and Theorem 3 to find a subsequence ${\bf u}_{n_{k}}$ that converges to a global solution of the Navier–Stokes equations. Theorem 1 is proven. $\diamond$

6 Solutions of the advection-diffusion problem with initial data in $L^{2}_{w_{\gamma}}$.

The proof of Theorem 1 on the Navier–Stokes problem can be easily adapted to the case of the advection-diffusion problem :

Theorem 4

Let $0<\gamma\leq 2$. Let $0<T<+\infty$. Let ${\bf u}_{0}$ be a divergence-free vector field such that ${\bf u}_{0}\in L^{2}_{w_{\gamma}}(\mathbb{R}^{3})$ and $\mathbb{F}$ be a tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that $\mathbb{F}\in L^{2}((0,T),L^{2}_{w_{\gamma}})$. Let ${\bf b}$ be a time-dependent divergence free vector-field (${\bf\nabla}\cdot{\bf b}=0$) such that ${\bf b}\in L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$.

Then the advection-diffusion problem

[TABLE]

has a solution ${\bf u}$ such that :

• $\bullet$

${\bf u}$* belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$*

• $\bullet$

the pressure $p$ is related to ${\bf u}$, ${\bf b}$ and $\mathbb{F}$ through the Riesz transforms $R_{i}=\frac{\partial_{i}}{\sqrt{-\Delta}}$ by the formula

[TABLE]

• $\bullet$

the map $t\in[0,T)\mapsto{\bf u}(t,.)$ is weakly continuous from $[0,T)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

there exists a non-negative locally finite measure $\mu$ on $(0,T)\times\mathbb{R}^{3}$ such that

[TABLE]

**Proof : **Again, we define $\phi_{R}(x)=\phi(\frac{x}{R})$, ${\bf u}_{0,R}=\mathbb{P}(\phi_{R}{\bf u}_{0})$ and $\mathbb{F}_{R}=\phi_{R}\mathbb{F}$. Moreover, we define ${\bf b}_{R}=\mathbb{P}(\phi_{R}{\bf b})$. We then solve the mollified problem

[TABLE]

for which we easily find a unique solution ${\bf u}_{R,\epsilon}$ in $L^{\infty}((0,T),L^{2})\cap L^{2}((0,T),\dot{H}^{1})$. Moreover, this solution belongs to $\mathcal{C}([0,T),L^{2})$.

Again, ${\bf u}_{R,\epsilon}$ fulfills the assumptions of Theorem 2 :

• $\bullet$

${\bf u}_{R,\epsilon}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{R,\epsilon}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

the map $t\in[0,T)\mapsto{\bf u}_{R,\epsilon}(t,.)$ is weakly continuous from $[0,T)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

on $(0,T)\times\mathbb{R}^{3}$, ${\bf u}_{R,\epsilon}$ fulfills the energy equality :

[TABLE]

with ${\bf b}_{R,\epsilon}={\bf b}_{R}*\theta_{\epsilon}$.

Thus, by Corollary 4 we know that,

[TABLE]

and

[TABLE]

where the constant $C_{\gamma}$ depends only on $\gamma$.

Moreover, we have that

[TABLE]

and

[TABLE]

Let $R_{n}\rightarrow+\infty$ and $\epsilon_{n}\rightarrow 0$. Let ${\bf u}_{0,n}={\bf u}_{0,T_{n}}$, $\mathbb{F}_{n}=\mathbb{F}_{R_{n}}$, ${\bf b}_{n}={\bf b}_{R_{n},\epsilon_{n}}$ and ${\bf u}_{n}={\bf u}_{R_{n},\epsilon_{n}}$. We may then apply Theorem 3, since ${\bf u}_{0,n}$ is strongly convergent to ${\bf u}_{0}$ in $L^{2}_{w_{\gamma}}$, $\mathbb{F}_{n}$ is strongly convergent to $\mathbb{F}$ in $L^{2}((0,T),L^{2}_{w_{\gamma}})$, and the sequence ${\bf b}_{n}$ is strongly convergent to ${\bf b}$ in $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$. Thus there exists $p$, ${\bf u}$ and an increasing sequence $(n_{k})_{k\in\mathbb{N}}$ with values in $\mathbb{N}$ such that

• $\bullet$

${\bf u}_{n_{k}}$ converges *-weakly to ${\bf u}$ in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$, ${\bf\nabla}{\bf u}_{n_{k}}$ converges weakly to ${\bf\nabla}{\bf u}$ in $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

$p_{n_{k}}$ converges weakly to $p$ in $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})+L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

${\bf u}_{n_{k}}$ converges strongly to ${\bf u}$ in $L^{2}_{\rm loc}([0,T)\times\mathbb{R}^{3})$.

We then easily finish the proof. $\diamond$

7 Application to the study of $\lambda$-discretely self-similar solutions

We may now apply our results to the study of $\lambda$-discretely self-similar solutions for the Navier–Stokes equations.

Definition 1

Let ${\bf u}_{0}\in L^{2}_{\rm loc}(\mathbb{R}^{3})$. We say that ${\bf u}_{0}$ is a $\lambda$-discretely self-similar function ($\lambda$-DSS) if there exists $\lambda>1$ such that $\lambda{\bf u}_{0}(\lambda x)={\bf u}_{0}$.

A vector field ${\bf u}\in L^{2}_{\rm loc}([0,+\infty)\times\mathbb{R}^{3})$ is $\lambda$-DSS if there exists $\lambda>1$ such that $\lambda{\bf u}(\lambda^{2}t,\lambda x)={\bf u}(t,x)$.

A forcing tensor $\mathbb{F}\in L^{2}_{\rm loc}([0,+\infty)\times\mathbb{R}^{3})$ is $\lambda$-DSS if there exists $\lambda>1$ such that $\lambda^{2}\mathbb{F}(\lambda^{2}t,\lambda x)=\mathbb{F}(t,x)$.

We shall speak of self-similarity if ${\bf u}_{0}$, ${\bf u}$ or $\mathbb{F}$ are $\lambda$-DSS for every $\lambda>1$.

Examples :

• $\bullet$

Let $\gamma>1$ and $\lambda>1$. Then, for two positive constants $A_{\gamma,\lambda}$ and $B_{\gamma,\lambda}$, we have : if ${\bf u}_{0}\in L^{2}_{\rm loc}(\mathbb{R}^{3})$ is $\lambda$-DSS, then ${\bf u}_{0}\in L^{2}_{w_{\gamma}}$ and

[TABLE]

• $\bullet$

${\bf u}_{0}\in L^{2}_{\rm loc}$ is self-similar if and only if it is of the form ${\bf u}_{0}=\frac{{\bf w}_{0}(\frac{x}{|x|})}{|x|}$ with ${\bf w}_{0}\in L^{2}(S^{2})$.

• $\bullet$

$\mathbb{F}$ belongs to $L^{2}((0,+\infty),L^{2}_{w_{\gamma}})$ with $\gamma>1$ and is self-similar if and only if it is of the form $\mathbb{F}(t,x)=\frac{1}{t}\mathbb{F}_{0}(\frac{x}{\sqrt{t}})$ with $\int|\mathbb{F}_{0}(x)|^{2}\frac{1}{|x|}\,dx<+\infty$.

Proof :

• $\bullet$

If ${\bf u}_{0}$ is $\lambda$-DSS and if $k\in\mathbb{Z}$ we have

[TABLE]

with $\sum_{k\in\mathbb{Z}}\frac{\lambda^{k}}{(1+\lambda^{k})^{\gamma}}<+\infty$ for $\gamma>1$.

• $\bullet$

If ${\bf u}_{0}$ is self-similar, we have ${\bf u}_{0}(x)=\frac{1}{|x|}{\bf u}_{0}(\frac{x}{|x|})$. From this equality, we find that, for $\lambda>1$

[TABLE]

• $\bullet$

If $\mathbb{F}$ is self-similar, then it is of the form $\mathbb{F}(t,x)=\frac{1}{t}\mathbb{F}_{0}(\frac{x}{\sqrt{t}})$. Moreover, we have

[TABLE]

with $C_{\gamma}=\int_{0}^{+\infty}\frac{1}{(1+\sqrt{\theta})^{\gamma}}\frac{d\theta}{\sqrt{\theta}}<+\infty$. $\diamond$

In this section, we are going to give a new proof of the results of Chae and Wolf [3] and Bradshaw and Tsai [2] on the existence of $\lambda$-DSS solutions of the Navier–Stokes problem (and of Jia and Šverák [6] for self-similar solutions) :

Theorem 5

Let $4/3<\gamma\leq 2$ and $\lambda>1$. If ${\bf u}_{0}$ is a $\lambda$-DSS divergence-free vector field (such that ${\bf u}_{0}\in L^{2}_{w_{\gamma}}(\mathbb{R}^{3})$) and if $\mathbb{F}$ is a $\lambda$-DSS tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that $\mathbb{F}\in L^{2}_{\rm loc}([0,+\infty)\times\mathbb{R}^{3})$ , then the Navier–Stokes equations with initial value ${\bf u}_{0}$

[TABLE]

has a global weak solution ${\bf u}$ such that :

• •

${\bf u}$* is a $\lambda$-DSS vector field*

• $\bullet$

for every $0<T<+\infty$, ${\bf u}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

the map $t\in[0,+\infty)\mapsto{\bf u}(t,.)$ is weakly continuous from $[0,+\infty)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

• $\bullet$

the solution ${\bf u}$ is suitable : there exists a non-negative locally finite measure $\mu$ on $(0,+\infty)\times\mathbb{R}^{3}$ such that

[TABLE]

7.1 The linear problem.

Following Chae and Wolf, we consider an approximation of the problem that is consistent with the scaling properties of the equations : let $\theta$ be a non-negative and radially decreasing function in $\mathcal{D}(\mathbb{R}^{3})$ with $\int\theta\,dx=1$; We define $\theta_{\epsilon,t}(x)=\frac{1}{(\epsilon\sqrt{t})^{3}}\ \theta(\frac{x}{\epsilon\sqrt{t}})$. We then will study the “mollified” problem

[TABLE]

and begin with the linearized problem

[TABLE]

Lemma 11

Let $1<\gamma\leq 2$. Let $\lambda>1$ Let ${\bf u}_{0}$ be a $\lambda$-DSS divergence-free vector field such that ${\bf u}_{0}\in L^{2}_{w_{\gamma}}(\mathbb{R}^{3})$ and $\mathbb{F}$ be a $\lambda$-DSS tensor $\mathbb{F}(t,x)=\left(F_{i,j}(t,x)\right)_{1\leq i,j\leq 3}$ such that, for every $T>0$, $\mathbb{F}\in L^{2}((0,T),L^{2}_{w_{\gamma}})$. Let ${\bf b}$ be a $\lambda$-DSS time-dependent divergence free vector-field (${\bf\nabla}\cdot{\bf b}=0$) such that, for every $T>0$, ${\bf b}\in L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$.

Then the advection-diffusion problem

[TABLE]

has a unique solution ${\bf v}$ such that :

• $\bullet$

for every positive $T$, ${\bf v}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf v}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

the pressure $p$ is related to ${\bf v}$, ${\bf b}$ and $\mathbb{F}$ through the Riesz transforms $R_{i}=\frac{\partial_{i}}{\sqrt{-\Delta}}$ by the formula

[TABLE]

• $\bullet$

the map $t\in[0,+\infty)\mapsto{\bf v}(t,.)$ is weakly continuous from $[0,+\infty)$ to $L^{2}_{w_{\gamma}}$, and is strongly continuous at $t=0$ :

[TABLE]

This solution ${\bf v}$ is a $\lambda$-DSS vector field.

**Proof : **As we have $|{\bf b}(t,.)*\theta_{\epsilon,t}|\leq\mathcal{M}_{{\bf b}(t,.)}$ and thus

[TABLE]

we see that we can use Theorem 4 to get a solution ${\bf v}$ on $(0,T)$.

As clearly ${\bf b}*\theta_{\epsilon,t}$ belongs to $L^{2}_{t}L^{\infty}_{x}(K)$ for every compact subset $K$ of $(0,T)\times\mathbb{R}^{3}$, we can use Corollary 5 to see that ${\bf v}$ is unique.

Let ${\bf w}(t,x)=\frac{1}{\lambda}{\bf v}(\frac{t}{\lambda^{2}},\frac{x}{\lambda})$. As $b*\theta_{\epsilon,t}$ is still $\lambda$-DSS, we see that ${\bf w}$ is solution of $(LNS_{\epsilon})$ on $(0,T)$, so that ${\bf w}={\bf v}$. This means that ${\bf v}$ is $\lambda$-DSS. $\diamond$

7.2 The mollified Navier–Stokes equations.

The solution ${\bf v}$ provided by Lemma 11 belongs to $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$ (as ${\bf v}$ belongs to $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf v}$ belongs to $L^{2}((0,T),L^{2}_{w_{\gamma}})$). Thus we have a mapping $L_{\epsilon}:{\bf b}\mapsto{\bf v}$ which is defined from

[TABLE]

to $X_{T,\gamma}$ by $L_{\epsilon}({\bf b})={\bf v}$.

Lemma 12

For $4/3<\gamma$, $X_{T,\gamma}$ is a Banach space for the equivalent norms $\|{\bf b}\|_{L^{3}((0,T),L^{3}_{w_{3\gamma/2}})}$ and $\|{\bf b}\|_{L^{3}((0,T/{\lambda^{2}}),\times B(0,\frac{1}{\lambda}))}$.

**Proof : **We have

[TABLE]

and , for $k\in\mathbb{N}$,

[TABLE]

We may conclude, since for $\gamma>4/3$ we have $\sum_{k\in\mathbb{N}}\lambda^{k(2-\frac{3\gamma}{2})}<+\infty$.

Lemma 13

For $4/3<\gamma\leq 2$, the mapping $L_{\epsilon}$ is continuous and compact on $X_{T,\gamma}$.

**Proof : **Let ${\bf b}_{n}$ be a bounded sequence in $X_{T,\gamma}$ and let ${\bf v}_{n}=L_{\epsilon}({\bf b}_{n})$. We remark that the sequence ${\bf b}_{n}(t,.)*\theta_{\epsilon,t}$ is bounded in $X_{T,\gamma}$. Thus, by Theorem 2 and Corollary 4, the sequence ${\bf v}_{n}$ is bounded in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf v}_{n}$ is bounded in $L^{2}((0,T),L^{2}_{w_{\gamma}})$.

We now use Theorem 3 and get that then there exists $q_{\infty}$, ${\bf v}_{\infty}$, ${\bf B}_{\infty}$ and an increasing sequence $(n_{k})_{k\in\mathbb{N}}$ with values in $\mathbb{N}$ such that

• $\bullet$

${\bf v}_{n_{k}}$ converges *-weakly to ${\bf v}_{\infty}$ in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$, ${\bf\nabla}{\bf v}_{n_{k}}$ converges weakly to ${\bf\nabla}{\bf v}_{\infty}$ in $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

${\bf b}_{n_{k}}*\theta_{\epsilon,t}$ converges weakly to ${\bf B}_{\infty}$ in $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$, ,

• $\bullet$

the associated pressures $q_{n_{k}}$ converge weakly to $q_{\infty}$ in $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})+L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

${\bf v}_{n_{k}}$ converges strongly to ${\bf v}_{\infty}$ in $L^{2}_{\rm loc}([0,T)\times\mathbb{R}^{3})$ : for every $T_{0}\in(0,T)$ and every $R>0$, we have

[TABLE]

As $\sqrt{w_{\gamma}}{\bf v}_{n}$ is bounded in $L^{\infty}((0,T),L^{2})$ and in $L^{2}((0,T),L^{6})$, it is bounded in $L^{10/3}((0,T)\times\mathbb{R}^{3})$. The strong convergence of ${\bf v}_{n_{k}}$ in $L^{2}_{\rm loc}([0,T)\times\mathbb{R}^{3})$ then implies the strong convergence of ${\bf v}_{n_{k}}$ in $L^{3}_{\rm loc}((0,T)\times\mathbb{R}^{3})$.

Moreover, ${\bf v}_{\infty}$ is still $\lambda$-DSS (a property that is stable under weak limits).We find that ${\bf v}_{\infty}\in X_{T,\gamma}$ and that

[TABLE]

This proves that $L_{\epsilon}$ is compact.

If we assume moreover that ${\bf b}_{n}$ is convergent to ${\bf b}_{\infty}$ in $X_{T,\gamma}$, then necessarily we have ${\bf B}_{\infty}={\bf b}_{\infty}*\theta_{\epsilon,t}$, and ${\bf v}_{\infty}=L_{\epsilon}({\bf b}_{\infty})$. Thus, the relatively compact sequence ${\bf v}_{n}$ can have only one limit point; thus it must be convergent. This proves that $L_{\epsilon}$ is continuous. $\diamond$

Lemma 14

Let $4/3<\gamma\leq 2$. If, for some $\mu\in[0,1]$, ${\bf v}$ is a solution of ${\bf v}=\mu L_{\epsilon}({\bf v})$ then

[TABLE]

where the constant $C_{{\bf u}_{0},\mathbb{F},\gamma,T}$ depends only on ${\bf u}_{0}$, $\mathbb{F}$, $\gamma$ and $T$ (but not on $\mu$ nor on $\epsilon$).

**Proof : **We have ${\bf v}=\mu{\bf w}$; with

[TABLE]

Multiplying by $\mu$, we find that

[TABLE]

We then use Corollary 6. We choose $T_{0}\in(0,T)$ such that

[TABLE]

Then, as

[TABLE]

we know that

[TABLE]

and

[TABLE]

In particular, we have

[TABLE]

As ${\bf v}$ is $\lambda$-DSS, we can go back from $T_{0}$ to $T$. $\diamond$

Lemma 15

Let $4/3<\gamma\leq 2$. There is at least one solution ${\bf u}_{\epsilon}$ of the equation ${\bf u}_{\epsilon}=L_{\epsilon}({\bf u}_{\epsilon})$.

**Proof : **Obvious due to the Leray–Schauder principle (and the Schaefer theorem), since $L_{\epsilon}$ is continuous and compact and since we have uniform a priori estimates for the fixed points of $\mu L_{\epsilon}$ for $0\leq\mu\leq 1$. $\diamond$

7.3 Proof of Theorem 5.

We may now finish the proof of Theorem 5. We consider the solutions ${\bf u}_{\epsilon}$ of ${\bf u}_{\epsilon}=L_{\epsilon}({\bf u}_{\epsilon})$.

By Lemma 14, ${\bf u}_{\epsilon}$ is bounded in $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$, and so is ${\bf u}_{\epsilon}*\theta_{\epsilon,t}$. We then know, by Theorem 2 and Corollary 4, that the familly ${\bf u}_{\epsilon}$ is bounded in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$ and ${\bf\nabla}{\bf u}_{\epsilon}$ is bounded in $L^{2}((0,T),L^{2}_{w_{\gamma}})$.

We now use Theorem 3 and get that then there exists $p$, ${\bf u}$, ${\bf B}$ and a decreasing sequence $(\epsilon_{k})_{k\in\mathbb{N}}$ (converging to [math]) with values in $(0,+\infty)$ such that

• $\bullet$

${\bf u}_{\epsilon_{k}}$ converges *-weakly to ${\bf u}$ in $L^{\infty}((0,T),L^{2}_{w_{\gamma}})$, ${\bf\nabla}{\bf u}_{\epsilon_{k}}$ converges weakly to ${\bf\nabla}{\bf u}$ in $L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

${\bf u}_{\epsilon_{k}}*\theta_{\epsilon_{k},t}$ converges weakly to ${\bf B}$ in $L^{3}((0,T),L^{3}_{w_{3\gamma/2}})$

• $\bullet$

the associated pressures $p_{\epsilon_{k}}$ converge weakly to $p$ in $L^{3}((0,T),L^{6/5}_{w_{\frac{6\gamma}{5}}})+L^{2}((0,T),L^{2}_{w_{\gamma}})$

• $\bullet$

${\bf u}_{\epsilon_{k}}$ converges strongly to ${\bf u}$ in $L^{2}_{\rm loc}([0,T)\times\mathbb{R}^{3})$.

Moreover we easily see that ${\bf B}={\bf u}$. Indeed, we have that ${\bf u}*\theta_{\epsilon,t}$ converges strongly in $L^{2}_{\rm loc}((0,T)\times\mathbb{R}^{3})$ as $\epsilon$ goes to [math] (since it is bounded by $\mathcal{M}_{\bf u}$ and converges, for each fixed $t$, strongly in $L^{2}_{\rm loc}(\mathbb{R}^{3})$); moreover, we have $|({\bf u}-{\bf u}_{\epsilon})*\theta_{\epsilon,t}|\leq\mathcal{M}_{{\bf u}-{\bf u}_{\epsilon}}$, so that the strong convergence of ${\bf u}_{\epsilon_{k}}$ to ${\bf u}$ is kept by convolution with $\theta_{\epsilon,t}$ as far as we work on compact subsets of $(0,T)\times\mathbb{R}^{3}$ (and thus don’t allow $t$ to go to [math]).

Thus, Theorem 5 is proven. $\diamond$