Uniqueness of weak solution to the generalized magneto-hydrodynamic system

Abstract

We study uniqueness of weak solution for the generalized incompressible magneto-hydrodynamic (GMHD) system with suitable $\beta$, and we prove that the weak solutions are unique in the class $L^{\frac{2\beta }{2\beta -1+r}}(0,T;B^{r}_{\infty ,\infty })$ with $r\in (1-2\beta ,1]$.

1 Introduction

In this paper, we consider the following generalized incompressible magneto-hydrodynamic (GMHD) system in $\mathbb R ^{n}\times (0,T)$, $n\ge 3$:

$$\begin{aligned}&u_{t}+(-\Delta )^{\beta } u +(u\cdot \nabla )u-(b\cdot \nabla )b+\nabla {P}=0,\quad \quad (x,t)\in \mathbb R ^{n}\times (0,T),\end{aligned}$$

(1.1)

$$\begin{aligned}&b_{t}+(-\Delta )^{\beta } b +(u\cdot \nabla )b-(b\cdot \nabla )u=0,\quad \quad (x,t)\in \mathbb R ^{n}\times (0,T),\end{aligned}$$

(1.2)

$$\begin{aligned}&\mathrm div u=0, \quad \mathrm div b=0,\quad \quad (x,t)\in \mathbb R ^{n}\times (0,T) \end{aligned}$$

(1.3)

with the initial conditions

$$\begin{aligned}&u(x,0)=u_{0}(x), \quad b(x,0)=b_{0}(x),\quad \quad x\in \mathbb R ^n, \end{aligned}$$

(1.4)

where $u=u(x,t)=(u^{1}(x,t),\ldots ,u^{n}(x,t))$, $b=b(x,t)=(b^{1}(x,t),\ldots ,b^{n}(x,t))$ and $P=P(x,t)$ stand for the fluid velocity, the magnetic field, and the total kinetic pressure, respectively, and $\beta \in (\frac{1}{2},1)$. The fractional Laplace operator $(-\Delta )^{\beta }$ with respect to space variable $x$ is a Riesz potential operator defined as usual through Fourier transform as $\mathcal F ((-\Delta )^{\beta }f)(\xi )=|\xi |^{2\beta }\mathcal F f(\xi )$, where $\mathcal F f(\xi )=\widehat{f}(\xi )=\frac{1}{\sqrt{2\pi }^{n}}\int _\mathbb{R ^{n}}e^{-ix\xi }f(x)\text{ d}x$. The initial velocity field $u_{0}$ and the initial magnetic field $d_{0}$ satisfy $\mathrm div u_{0}=0$ and $\mathrm div b_{0}=0$.

The GMHD system (1.1)–(1.4) describes the macroscopic behavior of the electrically conducting incompressible fluids in a magnetic field; it includes the well-known Navier–Stokes equations ($\beta =1$, $b\equiv 0$) and the standard MHD equations ($\beta =1$). In the last several decades, there have been numerous studies on the GMHD problems by many physicists and mathematicians due to its physical importance, complexity and mathematical challenges, see for example, [3, 4, 7, 8, 13, 22, 25, 27–33] and the references therein. In [27], Wu proved that the system (1.1)–(1.4) has a global-in-time weak solution for any given divergence free initial value $(u_{0},b_{0})\in L^{2}(\mathbb R ^{n})$. Yuan [31] obtained the local-in-time existence and uniqueness of smooth solution for any given sufficient smooth initial data $(u_{0},b_{0})$. However, whether the global weak solution is regular and unique or the unique local smooth solution can exist globally is an outstanding challenge problem, just as the situation for the Navier–Stokes equations and the MHD equations. So, a lot of literatures are devoted to find regularity criteria for the local smooth solutions or to find the uniqueness criteria for the weak solutions for these equations, we refer the reader to see [2, 6, 7, 10, 11, 15, 17, 19–21, 23] for Navier–Stokes equations, and [4, 8, 13, 32, 34] for MHD equations.

For the Navier–Stokes equations, it is well-known the Leray-Hopf weak solutions are unique and regular in the class

$$\begin{aligned}&L^{q}(0,T;L^{p}) \quad \text{ with} \frac{2}{q}+\frac{n}{p}\le 1, n\le p\le \infty , \quad \text{ see}\,[10,12,15,21,23]\\&L^{q}(0,T;W^{1,p}) \quad \text{ with} \frac{2}{q}+\frac{n}{p}\le 2, \frac{n}{2}< p\le \infty , \quad \text{ see} \,[1,2]. \end{aligned}$$

Recently, by means of the Fourier localization technique and Bony’s paraproduct decomposition, Chen et al. [7], Chemin and Lerner [6], Lemarié [17] and May [20] extended the condition to

$$\begin{aligned}&L^{q}(0,T; B^{r}_{p,\infty }) \quad \text{ with} \frac{2}{q}\!+\!\frac{n}{p}\!=\!1+r, \frac{n}{1\!+ \!r}< p\le \infty , r\in (-1,1] \text{ and} (p,r) \ne (\infty ,1);\\&[7,20] \text{ or} C([0,T];B^{-1}_{\infty ,\infty })\quad \text{ see} [6,17]. \end{aligned}$$

The above regularity results have been proved to still hold for the MHD equations (see e.g., [8, 13, 25, 32]). For the 3D GMHD equations, Wu [27–30] obtained some regularity criteria only relying on the velocity $u$. Recent result obtained by Zhou [33] (see also [18]) states that if the weak solution $(u,b)$ satisfies

$$\begin{aligned} u\in L^{q}(0,T;L^{p})\quad \text{ with} \frac{2\beta }{q}+\frac{3}{p}\le 2\beta -1, \frac{3}{2\beta -1}<p\le \infty , \end{aligned}$$

then $(u,b)$ is regular on $(0,T]$. Yuan in [31] extended this result to the case

$$\begin{aligned}&u\in L^{q}(0,T; B^{s}_{p,\infty })\quad \text{ with} \frac{2\beta }{q}+\frac{3}{p}\le 2\beta -1+s,\\&\frac{3}{2\beta -1+s}<p\le \infty .-1<s\le 1 \text{ and} (p,s)\ne (\infty ,1). \end{aligned}$$

The purpose of this paper is to uniqueness conditions of weak solutions for the GMHD system (1.1)–(1.4) in some Besov spaces. The tools we will use are mainly the Littlewood-Paley theory, the Bonys paraproduct decomposition and the Chemin-Lerner spaces.

First we recall the definition of weak solutions to the GMHD system (1.1)–(1.4).

Definition 1.1

(weak solution) A measurable vector function $(u,b)$ is called a weak solution to the GMHD system on the interval $[0,T)$ with initial value $(u_{0},b_{0})\in L^{2}(\mathbb R ^{n})$, if it satisfies the following properties

• (i) $(u,b)\in L^{\infty }(0,T;L^{2}(\mathbb R ^{n}))\cap L^{2}(0,T;H^{\beta }(\mathbb R ^{n}))$.

• (ii) $\mathrm div u=\mathrm div b=0$ in the sense of distributions, that is,

  $$\begin{aligned} \int \limits _{0}^{T}\int \limits _\mathbb{R ^{n}} u\cdot \nabla \phi \text{ d}x\text{ d}t=\int \limits _{0}^{T}\int \limits _\mathbb{R ^{n}}b\cdot \nabla \phi \text{ d}x\text{ d}t=0, \end{aligned}$$

  for all $\phi \in C_{0}^{\infty }(\mathbb R ^{n}\times (0,T))$. $(u,b)$ verifies system (1.1)–(1.4) in the sense of distribution, that is,

  $$\begin{aligned}&\int \limits _{0}^{T}\int \limits _\mathbb{R ^{n}}(\partial _{t}\phi +(u\cdot \nabla )\phi )u\text{ d}x\text{ d}t+\int \limits _\mathbb{R ^{n}}u_{0}\phi (x,0)\text{ d}x=\int \limits _{0}^{T}\int \limits _\mathbb{R ^{n}} (u\Lambda ^{2\beta }\phi +(b\cdot \nabla )\phi b)\text{ d}x\text{ d}t,\\&\int \limits _{0}^{T}\int \limits _\mathbb{R ^{n}}(\partial _{t}\phi +(u\cdot \nabla )\phi )b\text{ d}x\text{ d}t+\int \limits _\mathbb{R ^{n}}b_{0}\phi (x,0)\text{ d}x=\int \limits _{0}^{T}\int \limits _\mathbb{R ^{n}} (b\Lambda ^{2\beta }\phi +(b\cdot \nabla )\phi u)\text{ d}x\text{ d}t, \end{aligned}$$

  for all $\phi \in C_{0}^{\infty }(\mathbb R ^{n}\times (0,T))$ with $\mathrm div \phi =0$, where $\Lambda =(-\Delta )^{\frac{1}{2}}$.

• (iii) $(u,b)$ satisfies the energy inequality, that is,

  $$\begin{aligned} \Vert u(t)\Vert _{L^{2}}^{2}+\Vert b(t)\Vert _{L^{2}}^{2}+2\int \limits _{0}^{t}(\Vert \Lambda ^{\beta }u(\tau )\Vert _{L^{2}}^{2}+ \Vert \Lambda ^{\beta }b(\tau )\Vert _{L^{2}}^{2})\text{ d}\tau \le \Vert u_{0}\Vert _{L^{2}}^{2}+\Vert b_{0}\Vert _{L^{2}}^{2}. \end{aligned}$$

The main results of this paper are as follows:

Theorem 1.2

Let $\beta \in (\frac{1}{2},1]$, $T>0$ and $(u_{0},b_{0})\in L^{2}(\mathbb R ^{n})$ with $\mathrm div u_{0}=\mathrm div b_{0}=0$. Let $(u_{1},b_{1})$ and $(u_{2},b_{2})$ be two weak solutions of the GMHD system (1.1)–(1.3) with the same initial data $(u_{0},b_{0})$ satisfying one of the following two conditions:

• (a) $(u_{1},b_{1})\in L^{\frac{2\beta }{2\beta -1+r_{1}}}(0,T;B^{r_{1}}_{\infty ,\infty })$ and $(u_{2},b_{2})\in L^{\frac{2\beta }{2\beta -1+r_{2}}}(0,T;B^{r_{2}}_{\infty ,\infty })$ for some $1-2\beta <r_{1},r_{2}<1$ such that $r_{1}+r_{2}>1-2\beta$.

• (b) $(u_{1},b_{1}),(u_{2},b_{2})\in L^{1}(0,T;B^{1}_{\infty ,\infty })$.

Then $(u_{1},b_{1})\equiv (u_{2},b_{2})$ a.e. on $\mathbb R ^{n}\times (0,T)$.

Theorem 1.3

Let $\beta \in [\frac{7}{8},1]$, $3\le n\le 4(2\beta -1)$, $T>0$ and $(u_{0},b_{0})\in L^{2}(\mathbb R ^{n})$ with $\mathrm div u_{0}=\mathrm div b_{0}=0$. Let $(u_{1},b_{1})$ and $(u_{2},b_{2})$ be two weak solutions of the GMHD system (1.1)–(1.3) with the same initial data $(u_{0},b_{0})$ satisfying

$$\begin{aligned} (u_{1},b_{1})\in L^{\frac{2\beta }{2\beta -1-r_{1}}}(0,T;B^{-r_{1}}_{\infty ,\infty }) \text{ and} (u_{2},b_{2})\in L^{\frac{2\beta }{2\beta -1-r_{2}}}(0,T; B^{-r_{2}}_{\infty ,\infty }) \end{aligned}$$

for some $0< r_{1},r_{2}\le 2\beta -1$. Then, $(u_{1},b_{1})\equiv (u_{2},b_{2})$ a.e. on $\mathbb R ^{n}\times (0,T)$.

Remarks

1. Due to the embedding $B^{r}_{p,\infty }\hookrightarrow B^{r-\frac{n}{p}}_{\infty ,\infty }$, Theorem 1.2 is still valid when the assumption ($a$) is substituted by the following assumption:

$$\begin{aligned} (u_{1},b_{1})\in L^{q_{1}}(0,T; B^{r_{1}}_{p_{1},\infty }) \text{ and} (u_{2},b_{2})\in L^{q_{2}}(0,T;B^{r_{2}}_{p_{2},\infty }), \end{aligned}$$

where $\frac{2\beta }{q_{i}}+\frac{n}{p_{i}}=2\beta -1+r_{i}$, $1-2\beta < r_{i}<1$, $\frac{n}{2\beta -1+r_{i}}<p_{i}\le \infty$, ($i=1,2$) and $r_{1}+r_{2}>1-2\beta$.

2. Due to the inequality (see [7] for its proofs)

$$\begin{aligned} \Vert f\Vert _{B^{1}_{\infty ,\infty }}\le C(\Vert f\Vert _{L^{2}}+\Vert \text{ curl} f\Vert _{B^{0}_{\infty ,\infty }}), \end{aligned}$$

the assertion related to the assumption ($b$) of Theorem 1.2 implies that the following Beale-Kato-Majda-type uniqueness criterion (see [1]) holds: if two weak solutions $(u_{1},b_{1})$ and $(u_{2},b_{2})$ of the GMHD system with the same initial data satisfy

$$\begin{aligned} (\text{ curl} u_{1},\text{ curl} b_{1}), (\text{ curl} u_{2},\text{ curl} b_{2})\in L^{1}(0,T;B^{0}_{ \infty ,\infty }). \end{aligned}$$

Then $(u_{1},b_{1})\equiv (u_{2},b_{2})$ a.e. on $\mathbb R ^{n}\times (0,T)$.

3. For the Navier–Stokes equations, Chemin [5] and Lemarié [17] obtained the uniqueness of weak solutions in $C(0,T;B^{-1}_{\infty ,\infty })$. It is a nature question whether the condition $r_{1}+r_{2}>1-2\beta$ in the assumption ($a$) can be removed. Theorem 1.3 shows that the answer to this question is affirmative in the case $\beta \in [\frac{7}{8},1]$ and $n\le 4(2\beta -1)$. However, it seems very difficult for the rest cases.

The rest part of this paper is organized as follows. In Sect. 2, we collect some preliminaries materials, including the Littlewood–Paley decomposition, the definition of Besov spaces and some useful lemmas. In Sect. 3, we give the proof of Theorem 1.2. In the last section, we give the proof of Theorem 1.3. Throughout this paper, we denote by $C$ an universal positive constant whose value may change from line to line, and the notation $A\lesssim B$ means that $A\le C B$. If $X$ is a Banach space, $T$ is a positive real number and $p\in [1, +\infty ]$, we denote by $L^{p}_{T}(X)$ or $L^{p}_{T}X$ the space $L^{p}(0,T;X)$. The norm of the space $X$ is denoted by $\Vert \cdot \Vert _{X}$.

2 Preliminaries

In this section, we are going to recall some basic facts on the Littlewood–Paley theory, the definition of Besov space and some useful lemmas. Part of the materials presented here can be found in [6, 9, 16, 24, 26, 30]. Let $\mathcal S (\mathbb R ^{n})$ be the space of Schwartz class of rapidly decreasing functions. Given $f\in \mathcal S (\mathbb R ^{n})$, the Fourier transform of $f$ is defined by

$$\begin{aligned} \mathcal F f=\widehat{f}:=\frac{1}{(2\pi )^{\frac{n}{2}}} \int \limits _\mathbb{R ^{n}}e^{-ix\cdot \xi }f(x)\text{ d}x. \end{aligned}$$

We choose two nonnegative functions $\chi ,\varphi \in \mathcal S (\mathbb R ^{n})$, respectively, support in $\mathcal B =\{\xi \in \mathbb R ^{n},|\xi |\le \frac{4}{3}\}$ and $\mathcal C =\{\xi \in \mathbb R ^{n},\frac{3}{4}\le |\xi |\le \frac{8}{3}\}$ such that

$$\begin{aligned}&\chi (\xi )+\sum \limits _{j\ge 0}\varphi (2^{-j}\xi )=1\quad \text{ for} \text{ all} \xi \in \mathbb R ^{n};\\&\sum \limits _{j\in \mathbb Z }\varphi (2^{-j}\xi )=1\quad \text{ for} \text{ all} \xi \in \mathbb R ^{n}\backslash \{0\}. \end{aligned}$$

Setting $\varphi _{j}=\varphi (2^{-j}\xi )$, then $\text{ supp}\varphi _{j}\cap \text{ supp}\varphi _{j^{\prime }}=\phi$ if $|j-j^{\prime }|\ge 2$ and $\text{ supp}\chi \cap \text{ supp}\varphi _{j}=\phi$ if $j\ge 1$. Let $h=\mathcal F ^{-1}\varphi$ and $\widetilde{h}=\mathcal F ^{-1}\chi$. Define the frequency localization operators

$$\begin{aligned}&\Delta _{j}f=0 \text{ for} j\le -2;\quad \Delta _{-1}f=S_{0}f=\chi (D)f;\\&\Delta _{j}f=\varphi (2^{-j}D)f=2^{nj}\int \limits _\mathbb{R ^{n}}h(2^{j}y)f(x-y)\text{ d}y,\quad \text{ for} j\ge 0;\\&S_{j}f = \chi (2^{-j}D)f=\sum \limits _{-1\le k\le j-1}\Delta _{k}f=2^{nj}\int \limits _\mathbb{R ^{n}}\widetilde{h}(2^{j}y)f(x-y)\text{ d}y. \end{aligned}$$

Informally, $\Delta _{j}=S_{j+1}-S_{j}$ is a frequency projection to the annulus $\{ |\xi |\approx 2^{j}\}$, while $S_{j}$ is the frequency projection to the ball $\{|\xi |\lesssim 2^{j}\}$. One easily verifies that with the above choice of $\varphi$,

$$\begin{aligned} \Delta _{j}\Delta _{k}f\equiv 0 \text{ if} |j-k|\ge 2 \text{ and} \Delta _{j}(S_{k-1}f\Delta _{k}f )\equiv 0 \text{ if} |j-k|\ge 5. \end{aligned}$$

We recall the Bony’s paraproduct decomposition. Let $u$ and $v$ be two temperate distributions, the paraproducts between $u$ and $v$ are defined by

$$\begin{aligned} T_{u}v:=\sum \limits _{j}S_{j-1}u\Delta _{j}v \text{ and} T_{v}u:=\sum \limits _{j}S_{j-1}v\Delta _{j} u. \end{aligned}$$

Define the remainder of the paraproduct $R(u,v)$ as

$$\begin{aligned} R(u,v):=\sum \limits _{|j-j^{\prime }|\le 1}\Delta _{j}u\Delta _{j^{\prime }}v. \end{aligned}$$

Then, we have the following Bony’s decomposition:

$$\begin{aligned} uv=T_{u}v+T_{v}u+R(u,v). \end{aligned}$$

(2.1)

We shall sometimes also use the following simplified decomposition

$$\begin{aligned} uv=T_{u}v+T^{\prime }_{v}u\quad \text{ with} \quad T^{\prime }_{v}u=T_{v}u+R(u,v)=\sum \limits _{j}S_{j+2}v\Delta _{j}u. \end{aligned}$$

Now we introduce the definition of inhomogeneous Besov spaces by means of the Littlewood–Paley projection $\Delta _{j}$ and $S_{j}$.

Definition 2.1

Let $r\in \mathbb R$, $1\le p,q\le \infty$, the inhomogeneous Besov space $B^{r}_{p,q}:=B^{r}_{p,q}(\mathbb R ^{n})$ is defined by

$$\begin{aligned} B^{r}_{p,q}(\mathbb R ^{n}):=\{f\in \mathcal S ^{\prime }(\mathbb R ^{n}); \Vert f\Vert _{B^{r}_{p,q}}<\infty \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{B^{r}_{p,q}}= \left\{ \begin{array}{l} \left(\sum \limits _{j=-1}^{\infty }2^{jrq}\Vert \Delta _{j}f\Vert _{L^{p}}^{q}\right)^{\frac{1}{q}} \quad \text{ for} q< \infty , \\ \sup _{j\ge -1} 2^{js}\Vert \Delta _{j}f\Vert _{L^{p}}\quad \quad \quad \text{ for} q=\infty . \end{array} \right. \end{aligned}$$

We introduce the well–known Bernstein’s Lemma; its proofs can be found in Chemin [5] or Danchin [9].

Lemma 2.2

(Bernstein’s Lemma) Let $1\le p\le q\le \infty$. Assume that $f\in L^{p}(\mathbb R ^{n})$, then there exist constants $C,C_{1}$ independent of $f,j$ such that

$$\begin{aligned}&\sup _{|\alpha |=k}\Vert \partial ^{\alpha }f\Vert _{L^{q}}\le C 2^{jk+nj(\frac{1}{p}-\frac{1}{q})}\Vert f\Vert _{L^{p}}\quad \text{ for} \mathrm supp \widehat{f}\subset \{|\xi |\lesssim 2^{j}\},\\&C_{1}^{-1} 2^{jk}\Vert f\Vert _{L^{p}}\le \sup _{|\alpha |=k}\Vert \partial ^{\alpha } f\Vert _{L^{p}}\le C_{1} 2^{jk}\Vert f\Vert _{L^{p}}\quad \text{ for} \mathrm supp \widehat{f}\subset \{|\xi |\approx 2^{j}\}. \end{aligned}$$

We also recall the definition of a class of spaces introduced by Chemin and Lerner [6].

Definition 2.3

Let $T>0$, $r\in \mathbb R$ and $p,q\in [1,+\infty ]$. The space $\widetilde{L}^{q}_{T}B^{r}_{p,\infty }$ is the space of distributions $u\in \mathcal S ^{\prime }(\mathbb R ^{n}\times \mathbb R )$ such that

$$\begin{aligned} \Vert u\Vert _{\widetilde{L}^{q}_{T}B^{r}_{p,\infty }}:=\sup _{j\ge -1}2^{rj}\Vert \Delta _{j}u\Vert _{L^{q}_{T}L^{p}_{x}}<\infty . \end{aligned}$$

The following proposition gives some other properties of the semigroup $(e^{-t(-\Delta )^{\beta }})_{t>0}$.

Proposition 2.4

Let $T>0$, $r,r_{1},r_{2}\in \mathbb R$ and $p,q\in [1,+\infty ]$, we have following assertions:

• (1) If $p\le q$, then the family $(t^{\frac{n}{2\beta }(\frac{1}{p}-\frac{1}{q})}e^{-t(-\Delta )^{\beta }} )_{t>0}$ is continuous from $L^{q}$ to $L^{p}$;

• (2) If $r_{1}<r_{2}$, then the family $(t^{\frac{r_{2}-r_{1}}{2\beta }}e^{-t(-\Delta )^{\beta }})_{0< t\le T}$ is continuous from $B^{r_{1}}_{p,\infty }$ to $B^{r_{2}}_{p,\infty }$;

• (3) The operator $e^{-t(-\Delta )^{\beta }}$ is continuous from $B^{r}_{p,\infty }$ to $\widetilde{L}^{q}_{T}B^{r+\frac{2\beta }{q}}_{p,\infty }$.

It is well-known that (see [17]) the semigroup $(e^{t\Delta })_{t\ge 0}$ can be used to characterize the inhomogeneous $B^{-r}_{p,\infty }$ with $r>0$ and $1\le p\le \infty$. This property still holds for the semigroup $(e^{-t(-\Delta )^{\beta }})_{t\ge 0}$; we have the following proposition in a particular case of this characterization.

Proposition 2.5

Let $1\le p\le \infty$ and $r>0$. Then, $f\in B^{-r}_{p,\infty }$ if and only if

$$\begin{aligned} \sup _{0<\tau <\delta }\tau ^{\frac{r}{2\beta }}\Vert e^{-\tau (-\Delta )^{\beta }}f\Vert _{L^{p}}<\infty \end{aligned}$$

(2.2)

for any $\delta >0$. Thus, the right side of (2.2) defines a equivalent norm of $B^{-r}_{p,\infty }$.

Proof

We first introduce the functions $\widetilde{\chi }(\xi )=\chi {(\frac{\xi }{2})}$ and $\widetilde{\varphi }(\xi )=\chi (\frac{\xi }{4})-\chi (4\xi )$ as well as the operator $\widetilde{S}_{j}$ and $\widetilde{\Delta }_{j}$ are defined by $\mathcal F (\widetilde{S}_{j}f)=\widetilde{\chi }({\frac{\xi }{2^{j}}})\mathcal F f$ and $\mathcal F (\widetilde{\Delta }_{j}f)=\widetilde{\varphi }(\frac{\xi }{2^{j}})\mathcal F f$ (Hence, we have $S_{j}=\widetilde{S}_{j}S_{j}$ and $\Delta {j}=\widetilde{\Delta }_{j}\Delta _{j}$).

We now first prove that

$$\begin{aligned} \sup _{0<t<\delta }t^{\frac{r}{2\beta }}\Vert e^{-t(-\Delta )^{\beta }}f\Vert _{L^{p}}\le C\sup _{j\ge -1}2^{jr}\Vert \Delta _{j}f\Vert _{L^{p}}. \end{aligned}$$

(2.3)

We may assume that $\delta =1$ and write $f=S_{0}f+\sum _{j\ge 0}\Delta _{j}f$ with $S_{0}f\in L^{p}$ and $\Vert \Delta _{j}f\Vert _{L^{p}}=2^{jr}\varepsilon _{j}$ with $(\varepsilon _{j})_{j\in \mathbb N }\in l^{\infty }$. We estimate the norm $\Vert t^{\frac{r}{2\beta }}e^{-t(-\Delta )^{\beta }}S_{0}f\Vert _{L^{p}}$ by

$$\begin{aligned} \Vert t^{\frac{r}{2\beta }}e^{-t(-\Delta )^{\beta }}S_{0}f\Vert _{L^{p}}&= \Vert t^{\frac{r}{2\beta }}e^{-t(-\Delta )^{\beta }}\widetilde{S}_{0}S_{0}f\Vert _{L^{p}}\nonumber \\&\le C t^{\frac{r}{2\beta }}\Vert \widetilde{S}_{0}S_{0}f\Vert _{L^{p}}\nonumber \\&\le C t^{\frac{r}{2\beta }}\Vert S_{0}f\Vert _{L^{p}}, \end{aligned}$$

where we have used the first property of Proposition 2.4 above and the fact that $\widetilde{S}_{0}$ is a convolution operator with an integrable kernel. Similarly, writing $\Delta _{j}f=\widetilde{\Delta }_{j}\Delta _{j}f$ and using the integrability of the kernel of the convolution operator $\widetilde{\Delta }_{0}$, we get

$$\begin{aligned} \Vert t^{\frac{r}{2\beta }}e^{-(-\Delta )^{\beta }}\Delta _{j}f\Vert _{L^{p}}&= \Vert t^{\frac{r}{2\beta }}e^{-(-\Delta )^{\beta }}\widetilde{\Delta }_{j}\Delta _{j}f\Vert _{L^{p}}\nonumber \\&\le C t^{\frac{r}{2\beta }}\Vert \widetilde{\Delta }_{j}\Delta _{j}f\Vert _{L^{p}}\le C t^{\frac{r}{2\beta }}\Vert \Delta _{j}f\Vert _{L^{p}}. \end{aligned}$$

Notice that $e^{-t(-\Delta )^{\beta }}=t^{-\frac{N}{2\beta }}(-t\Delta )^{\frac{N}{2\beta }}e^{-t(-\Delta )^{\beta }}(-\Delta )^{-\frac{N}{2\beta }}$ and the integrability of the kernel of $(-t\Delta )^{\beta }e^{-t(-\Delta )^{\beta }}$ and $(-\Delta )^{\frac{N}{2\beta }}\widetilde{\Delta }_{0}$, we get that

$$\begin{aligned} \Vert t^{\frac{r}{2\beta }}e^{-t(-\Delta )^{\beta }}\Delta _{j}f\Vert _{L^{p}}&= \Vert t^{\frac{r-N}{2\beta }}(-t\Delta )^{\beta }e^{-t(-\Delta )^{\beta }}(-\Delta )^{-\frac{N}{2\beta }}\widetilde{\Delta }_{j}\Delta _{j}f\Vert _{L^{p}}\nonumber \\&\le C_{N} t^{\frac{r-N}{2\beta }}\Vert (-\Delta )^{-\frac{N}{2\beta }}\widetilde{\Delta }_{j}\Delta _{j}f\Vert _{L^{p}}\nonumber \\&\le C_{N} t^{\frac{r-N}{2\beta }} 2^{-j\frac{N}{\beta }}\Vert \Delta _{j}f\Vert _{L^{p}}. \end{aligned}$$

Now, notice that $t\le 1$, we choose $j_{0}$ so that $\frac{1}{4}\le 2^{2\beta j_{0}}t\le 1$ and choose $N>r$, then we get

$$\begin{aligned}&\Vert r^{\frac{r}{2\beta }}e^{-t(-\Delta )^{\beta }}f\Vert _{L^{p}}\nonumber \\&\le C(2^{-rj_{0}}\Vert S_{0}f\Vert _{L^{p}}+\sum \limits _{0\le j\le j_{0}}2^{-rj_{0}}2^{jr}\varepsilon _{j})+C_{N}\sum \limits _{j\ge j_{0}}2^{j_{0}(r-N)}2^{jr}2^{-j\frac{N}{\beta }}\varepsilon _{j}\nonumber \\&\le C(2^{rj_{0}}+\sum \limits _{0\le j\le j_{0}}2^{(j-j_{0})r}+\sum \limits _{j\ge j_{0}}2^{(j-j_{0})(r-N)})\sup _{j\ge -1}2^{rj}\Vert \Delta _{j}f\Vert _{L^{p}}\nonumber \\&\le C \sup _{j\ge -1}2^{rj}\Vert \Delta _{j}f\Vert _{L^{p}}. \end{aligned}$$

Hence (2.3) is established. Now, in order to complete the proof, it is sufficient to prove

$$\begin{aligned} \sup _{j\ge -1}2^{-jr}\Vert \Delta _{j}f\Vert _{L^{p}}\le \sup _{0<t< \delta } t^{\frac{r}{2\beta }}\Vert e^{-t(-\Delta )^{\beta }}f\Vert _{L^{p}}. \end{aligned}$$

(2.4)

We assume again that $\delta =1$. It is easy to prove $S_{0}f\in L^{p}$ by writing $S_{0}f=e^{(-\Delta )^{\beta }}S_{0}e^{-(-\Delta )^{\beta }}f$ and using the integrability of the kernel of the convolution $e^{-(-\Delta )^{\beta }}S_{0}$. Similarly, we write $\Delta _{j}f=e^{t(-\Delta )^{\beta }}\Delta _{j}e^{-t(-\Delta )^{\beta }}f$. For $j\ge 0$, choose $0<t<\delta$ such that $\frac{1}{4}\le 2^{2\beta j}t\le 1$, the convolution operator $e^{t(-\Delta )^{\beta }}\Delta _{j}$ has an integrable kernel $K_{j,t}$ with $\Vert K_{j,t}\Vert _{L^{1}}\le C$, then we finally obtain

$$\begin{aligned} 2^{-jr}\Vert \Delta _{j}f\Vert _{L^{p}}&= 2^{-jr}\Vert e^{t(-\Delta )^{\beta }}\Delta _{j}e^{-t(-\Delta )^{\beta }}f\Vert _{L^{p}}\nonumber \\&\le C t^{\frac{r}{2\beta }}\Vert e^{-t(-\Delta )^{\beta }}f\Vert _{L^{p}}, \end{aligned}$$

which imply that (2.4). The proof of Proposition 2.5 is complete.$\square$

The following lemma is due to Lemarié [16, 17].

Lemma 2.6

Let $E_{2}$ be the closure of the test functions in the Morrey space $L^{2}_{uloc}$:

$$\begin{aligned} f\in E_{2}\Leftrightarrow \sup _{x_{0}\in \mathbb R ^{n}}\int \limits _{|x-x_{0}|<1}|f(x)|^{2}\mathrm{d}x<\infty \text{ and} \lim _{x_{0}\rightarrow \infty }\int \limits _{|x-x_{0}|<1}|f(x)|^{2}\mathrm{d}x=0. \end{aligned}$$

If $u\in L^{2}([0,a],E_{2})$, then the following assertions are equivalent:

• (1) $(u,b)$ is the solution of the Generalized Magneto-hydrodynamic system (1.1)–(1.4).

• (2) $(u,b)$ is the solution of the integral equations

$$\begin{aligned} \left\{ \begin{array}{l} u=e^{-t(-\Delta )^{\beta }}u_{0}-\int \limits _{0}^{t}e^{-(t-\tau )(-\Delta )^{\beta }}\mathbb P \nabla \cdot (u\otimes u+b\otimes b)(\tau )\text{ d}\tau ,\\ b=e^{-t(-\Delta )^{\beta }}b_{0}-2\int \limits _{0}^{t}e^{-(t-\tau )(-\Delta )^{\beta }}\mathbb P \nabla \cdot (u\otimes b)(\tau )\text{ d}\tau ,\\ \mathrm div u_{0}=0, \quad \mathrm div b_{0}=0. \end{array} \right. \end{aligned}$$

(2.5)

We end the section by the following propositions gathering some simple and useful properties of Besov spaces and Chemin–Lerner spaces (see [5, 6, 9]).

Proposition 2.7

Let $T>0$, $r\in \mathbb R$ and $p,q\in [1,+\infty ]$. The following assertions are true:

• (1) $\ L^{q}_{T}B^{r}_{p,\infty }\subset \widetilde{L}^{q}_{T}B^{r}_{p,\infty }$, $L^{\infty }_{T}B^{r}_{p,\infty }=\widetilde{L}^{\infty }_{T}B^{r}_{p,\infty }$;

• (2) The operator $\mathbb P _{ij}\frac{\partial }{\partial _{i}}$ maps continuously from $B^{r}_{p,\infty }$ (respectively, $L^{q}_{T}B^{r}_{p,\infty }$) to $B^{r-1}_{p,\infty }$ (respectively, $L^{q}_{T}B^{r-1}_{p,\infty }$);

• (3) For any $m\in [p,\infty ]$, we have

  $$\begin{aligned} B^{r}_{p,\infty }\subset B^{r+n(\frac{1}{m}-\frac{1}{p})}_{m,\infty },\quad \widetilde{L}^{q}_{T}B^{r}_{p,\infty }\subset \widetilde{L}^{q}_{T}B^{r+n(\frac{1}{m}-\frac{1}{p})}_{m,\infty }. \end{aligned}$$

Proposition 2.8

Let $T>0$, $0<r_{1}<r_{2}<+\infty$ and $p,q, p_{1},q_{1},p_{2},q_{2}\in [0, +\infty ]$ such that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $\frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$. Then, the following assertions are true:

• (1) The paraproduct operator of Bony $T_{u}v$ and the remainder operator $R(u,v)$ are continuous from $B^{-r_{1}}_{p_{1},\infty }\times B^{r_{2}}_{p_{2},\infty }$ to $B^{r_{2}-r_{1}}_{p,\infty }$;

• (2) The paraproduct operator of Bony $T_{u}v$ and the remainder operator $R(u,v)$ are continuous from $\widetilde{L}^{q_{1}}_{T}B^{-r_{1}}_{p_{1},\infty }\times \widetilde{L}^{q_{2}}_{T}B^{r_{2}}_{p_{2},\infty }$ to $\widetilde{L}^{q}_{T}B^{r_{2}-r_{1}}_{p,\infty }$, as well as continuous from $L^{q_{1}}_{T}L^{p_{1}}_{x}\times \widetilde{L}^{q_{2}}_{T}B^{r_{2}}_{p_{2},\infty }$ to $\widetilde{L}^{p}_{T}B^{r_{2}}_{p,\infty }$.

3 Proof of Theorem 1.2

In this section, we give the proof of Theorem 1.2. Assume that $(u_{1},b_{1})$ and $(u_{2},b_{2})$ are two weak solutions of system (1.1)–(1.4) on (0,T) with the same initial data $(u_{0},b_{0})$. Let $u=u_{1}-u_{2}$, $b=b_{1}-b_{2}$, then $(u,b)$ satisfies the following two equations in the sense of distribution

$$\begin{aligned}&\partial _{t}u+(-\Delta )^{\beta }u+u\cdot \nabla u_{1}+u_{2}\cdot \nabla u-b\cdot \nabla b_{1}-b_{2}\cdot \nabla b+\nabla \widetilde{P}=0;\end{aligned}$$

(3.1)

$$\begin{aligned}&\partial _{t}b+(-\Delta )^{\beta }b-b\cdot \nabla u_{1}-b_{2}\cdot \nabla u+u\cdot \nabla b_{1}+u_{2}\cdot \nabla b=0, \end{aligned}$$

(3.2)

for some pressure $\widetilde{P}$.

Multiplying $\Delta _{j}$ of (3.1) with $\Delta _{j}u$, and integrating on space variable, we get by Lemma 2.2 for $j\ge -1$ that

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \Delta _{j} u\Vert _{L^{2}}^{2}+a_{j}2^{2j\beta }\Vert \Delta _{j} u\Vert _{L^{2}}^{2}\nonumber \\&= -\langle \Delta _{j}(u\cdot \nabla u_{1}),\Delta _{j}u \rangle - \langle \Delta _{j}(u_{2}\cdot \nabla u),\Delta _{j}u \rangle +\langle \Delta _{j}(b\cdot \nabla b_{1}),\Delta _{j}u \rangle +\langle \Delta _{j}(b_{2}\cdot \nabla b),\nonumber \\&\Delta _{j}u \rangle \nonumber \\&= -\langle \Delta _{j}(u\cdot \nabla u_{1}),\Delta _{j}u \rangle + \langle [u_{2},\Delta _{j}]\nabla u,\Delta _{j}u \rangle +\langle \Delta _{j}(b\cdot \nabla b_{1}),\Delta _{j}u \rangle +\langle \Delta _{j}(b_{2}\cdot \nabla b),\nonumber \\&\Delta _{j}u \rangle \nonumber \\&\equiv \, A_{1}+A_{2}+A_{3}+A_{4}, \end{aligned}$$

(3.3)

where $[A,B]:= AB-BA$, $a_{-1}=0$ and $a_{j}=1$ for $j\ge 0$. Here we have used the fact $\langle u_{2}\cdot \nabla \Delta _{j}u,\Delta _{j}u\rangle =0$. Similarly, multiplying $\Delta _{j}$ of (3.2) with $\Delta _{j} b$ and integrating on space variable, then

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \Delta _{j} b\Vert _{L^{2}}^{2}+a_{j}2^{2j\beta }\Vert \Delta _{j} b\Vert _{L^{2}}^{2}\nonumber \\&= -\langle \Delta _{j}(u\cdot \nabla b_{1}),\Delta _{j}b \rangle - \langle \Delta _{j}(u_{2}\cdot \nabla b),\Delta _{j}b \rangle +\langle \Delta _{j}(b\cdot \nabla u_{1}),\Delta _{j}b \rangle +\langle \Delta _{j}(b_{2}\cdot \nabla u),\nonumber \\&\Delta _{j}b \rangle \nonumber \\&= -\langle \Delta _{j}(u\cdot \nabla b_{1}),\Delta _{j}b \rangle + \langle [u_{2},\Delta _{j}]\nabla b,\Delta _{j}b \rangle +\langle \Delta _{j}(b\cdot \nabla u_{1}),\Delta _{j}b \rangle +\langle \Delta _{j}(b_{2}\cdot \nabla u),\nonumber \\&\Delta _{j}b \rangle \nonumber \\&\equiv \, B_{1}+B_{2}+B_{3}+B_{4}, \end{aligned}$$

(3.4)

where we have used the fact $\langle u_{2}\cdot \nabla \Delta _{j}b,\Delta _{j}b\rangle =0$.

Now, we will estimate $A_{k},B_{k}(k=1,2,3,4)$ term by term. Using the Bony’s decomposition (2.1), we have

$$\begin{aligned} \langle \Delta _{j}(u\cdot \nabla u_{1}),\Delta _{j}u\rangle&= \langle \Delta _{j}(T_{u^{i}}\partial _{i}u_{1})+\Delta _{j}T_{\partial _{i}u_{1}}u^{i}+\Delta _{j}R(u^{i},\partial _{i}u_{1}),\Delta _{j}u\rangle \nonumber \\&= A_{11}+A_{12}+A_{13}. \end{aligned}$$

(3.5)

Considering the support of Fourier transform of the term $T_{u^{i}}\partial _{i}u_{1}$ and the definition of $\Delta _{j}$, we have

$$\begin{aligned} \Delta _{j}(T_{u^{i}}\partial _{i}u_{1})=\sum \limits _{|j^{\prime }-j|\le 4}\Delta _{j}(S_{j^{\prime }-1}u^{i}\partial _{i}\Delta _{j^{\prime }}u_{1}). \end{aligned}$$

Hence, by using Hölder’s inequality and Bernstein’s Lemma, $A_{11}$ can be estimated as

$$\begin{aligned} A_{11}&\lesssim \Vert \Delta _{j}(T_{u^{i}}\partial _{i}u_{1})\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim \sum \limits _{|j^{\prime }-j|\le 4}2^{j^{\prime }}\sum \limits _{j^{\prime \prime }\le j^{\prime }-2}\Vert \Delta _{j^{\prime \prime }}u\Vert _{L^{2}}\Vert \Delta _{j^{\prime }}u_{1}\Vert _{L^{\infty }}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim 2^{j}(1-r_{1})\Vert u_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j+2}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}. \end{aligned}$$

(3.6)

Similarly, there holds

$$\begin{aligned} \Delta _{j}(T_{\partial _{i}u_{1}}u^{i})=\sum \limits _{|j^{\prime }-j|\le 4}\Delta _{j}(S_{j^{\prime }-1}(\partial _{i}u_{1})\Delta _{j^{\prime }}u^{i}). \end{aligned}$$

(3.7)

The equality (3.7) as well as Hölder’s inequality and Bernstein’s Lemma implies that

$$\begin{aligned} A_{12}&\lesssim \Vert \Delta _{j}(T_{\partial _{i}u_{1}}u^{i})\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim \sum \limits _{|j^{\prime }-j|\le 4}\sum \limits _{j^{\prime \prime }\le j^{\prime }-2} 2^{j^{\prime \prime }}\Vert \Delta _{j^{\prime \prime }}u_{1}\Vert _{L^{\infty }}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim 2^{j(1-r_{1})}\Vert u_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{|j^{\prime }-j|\le 4}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j} u\Vert _{L^{2}}. \end{aligned}$$

(3.8)

For the remainder term $R(u^{i},\partial _{i}u_{1})$, we have

$$\begin{aligned} \Delta _{j}R(u^{i},\partial _{i}u_{1})&= \sum \limits _{j^{\prime },j^{\prime \prime }\ge j-3;|j^{\prime }-j^{\prime \prime }|\le 1}\Delta _{j}(\Delta _{j^{\prime }}u^{i}\Delta _{j^{\prime \prime }}\partial _{i}u_{1})\nonumber \\&= \sum \limits _{j^{\prime },j^{\prime \prime }\ge j-3;|j^{\prime }-j^{\prime \prime }|\le 1}\partial _{i}\Delta _{j}(\Delta _{j^{\prime }}u^{i}\Delta _{j^{\prime \prime }}u_{1}), \end{aligned}$$

(3.9)

where we have use the divergence free condition $\mathrm div u=0$. Hence,

$$\begin{aligned} A_{13}&\lesssim \Vert \Delta _{j}R(u^{i},\partial _{i}u_{1})\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim \sum \limits _{j^{\prime },j^{\prime \prime }\ge j-3;|j^{\prime }-j^{\prime \prime }|\le 1} 2^{j}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j^{\prime \prime }}u_{1}\Vert _{L^{\infty }}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim 2^{j}\Vert u_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j-3}2^{-j^{\prime }r_{1}}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}. \end{aligned}$$

(3.10)

Combining (3.5), (3.6), (3.8) and (3.10) together, it follows that

$$\begin{aligned} A_{1}&\lesssim 2^{j(1-r_{1})}\Vert u_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&+\,2^{j}\Vert u_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j}2^{-j^{\prime }r_{1}}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}. \end{aligned}$$

(3.11)

Similarly, we have

$$\begin{aligned} A_{3}&\lesssim 2^{j(1-r_{1})}\Vert b_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}\Vert \Delta _{j^{\prime }}b\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\!\nonumber \\&+\,2^{j}\Vert b_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j}2^{-j^{\prime }r_{1}}\Vert \Delta _{j^{\prime }}b\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}};\end{aligned}$$

(3.12)

$$\begin{aligned} B_{1}&\lesssim 2^{j(1-r_{1})}\Vert b_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}b\Vert _{L^{2}}\!\nonumber \\&+\,2^{j}\Vert b_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j}2^{-j^{\prime }r_{1}}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}b\Vert _{L^{2}};\end{aligned}$$

(3.13)

$$\begin{aligned} B_{3}&\lesssim 2^{j(1-r_{1})}\Vert u_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}\Vert \Delta _{j^{\prime }}b\Vert _{L^{2}}\Vert \Delta _{j}b\Vert _{L^{2}}\!\nonumber \\&+\,2^{j}\Vert u_{1}\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j}2^{-j^{\prime }r_{1}}\Vert \Delta _{j^{\prime }}b\Vert _{L^{2}}\Vert \Delta _{j}b\Vert _{L^{2}}, \end{aligned}$$

(3.14)

and similar as (3.6), (3.8) and (3.10) with few changes, we obtain

$$\begin{aligned} A_{4}&\lesssim 2^{-jr_{2}}\Vert b_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}2^{j^{\prime }}\Vert \Delta _{j^{\prime }}b\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\!\nonumber \\&+\,2^{j}\Vert b_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j}2^{-j^{\prime }r_{2}}\Vert \Delta _{j^{\prime }}b\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}};\end{aligned}$$

(3.15)

$$\begin{aligned} B_{4}&\lesssim 2^{-jr_{2}}\Vert b_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}2^{j^{\prime }}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}b\Vert _{L^{2}}\!\nonumber \\&+\,2^{j}\Vert b_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j}2^{-j^{\prime }r_{2}}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}b\Vert _{L^{2}}. \end{aligned}$$

(3.16)

To estimate the terms $A_{2}$ and $B_{2}$. By using Bony’s decomposition again, there holds

$$\begin{aligned}{}[f,\Delta _{j}]g&= f\cdot \nabla \Delta _{j}g-\Delta _{j}(f\cdot \nabla g)\nonumber \\&= T_{f^{i}}\Delta _{j}\partial _{i}g+T_{\Delta _{j}\partial _{i}g}f^{i}+R(f^{i},\partial _{i}\Delta _{j}g)-\Delta _{j} (T_{f^{i}}\partial _{i}g)-\Delta _{j}(T_{\partial _{i}g}f^{i})\nonumber \\&-\Delta _{j}R(f^{i},\partial _{i}g)\nonumber \\&= [T_{f^{i}},\Delta _{j}]\partial _{i}g+T^{\prime }_{\Delta _{j}\partial _{i}g}f^{i}-\Delta _{j}(T_{\partial _{i}g}f^{i}) -\Delta _{j}R(f^{i},\partial _{i}g). \end{aligned}$$

Hence, we can rewrite $A_{2}$ as

$$\begin{aligned} A_{2}&= \langle [T_{u_{2}^{i}},\Delta _{j}]\partial _{i}u,\Delta _{j}u\rangle + \langle T^{\prime }_{\Delta _{j}\partial _{i}u}u_{2}^{i},\Delta _{j}u \rangle -\langle \Delta _{j}(T_{\partial _{i}u}u_{2}^{i}),\Delta _{j}u \rangle \nonumber \\&-\langle \Delta _{j}R(u_{2}^{i},\partial _{i}u),\Delta _{j}u \rangle \nonumber \\&\equiv \, A_{21}+A_{22}+A_{23}+A_{24}. \end{aligned}$$

(3.17)

Making use of the definition of $\Delta _{j}$, we have

$$\begin{aligned}{}[T_{u_{2}^{i}},\Delta _{j}]\partial _{i}u&= \sum \limits _{|j^{\prime }-j|\le 4}\left[S_{j^{\prime }-1}u_{2}^{i},\Delta _{j}\right]\partial _{i}\Delta _{j^{\prime }}u\nonumber \\&= \sum \limits _{|j^{\prime }-j|\le 4} 2^{nj}\int \limits _\mathbb{R ^{n}}h(2^{j}(x-y))\left(S_{j^{\prime }-1}u_{2}^{i}(x)-S_{j^{\prime }-1}u_{2}^{i}(y)\right) \partial _{i}\Delta _{j^{\prime }}u(y)\text{ d}y\nonumber \\&= \sum \limits _{|j^{\prime }-j|\le 4} 2^{n+1}\int \limits _\mathbb{R ^{n}}\int \limits _{0}^{1}y\cdot \nabla S_{j^{\prime }-1}u_{2}^{i}(x-\tau y)\text{ d}\tau \partial _{i}h(2^{j}y)\Delta _{j^{\prime }}g(x-y)\text{ d}y, \end{aligned}$$

from which and Hölder’s inequality, it follows that

$$\begin{aligned} A_{21}&\lesssim \Vert [T_{u_{2}^{i}},\Delta _{j}]\partial _{i}u\Vert _{L^{2}}\Vert \partial _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim \sum \limits _{|j^{\prime }-j|\le 4}\Vert \nabla S_{j^{\prime }-1} u_{2}\Vert _{L^{\infty }}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim 2^{-jr_{2}}\Vert u_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{|j^{\prime }-j|\le 4}2^{j^{\prime }}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j} u\Vert _{L^{2}}. \end{aligned}$$

(3.18)

Considering the support of Fourier transform of the term $T^{\prime }_{\Delta _{j}\partial _{i}u}u^{i}_{2}$ and the definition of $\Delta _{j}$, we have

$$\begin{aligned} \langle T^{\prime }_{\Delta _{j}\partial _{i}u}u_{2}^{i}, \Delta _{j}u\rangle&= \left\langle \sum \limits _{j^{\prime }\ge j-2} S_{j^{\prime }+2}\Delta _{j}\partial _{i}u \Delta _{j^{\prime }}u_{2}^{i}, \Delta _{j} u \right\rangle \nonumber \\&= \left\langle \sum \limits _{ j-2\le j^{\prime }\le j} S_{j^{\prime }+2}\Delta _{j}\partial _{i}u \Delta _{j^{\prime }}u_{2}^{i}, \Delta _{j} u \right\rangle , \end{aligned}$$

where we have used the facts that $S_{j^{\prime }+2}\Delta _{j}u=\Delta _{j}u$ for $j^{\prime }>j$ and $\langle \Delta _{j}\partial _{i}u\Delta _{j^{\prime }}u_{2}^{i},\Delta _{j}u \rangle =0$. It follows from the above equality that

$$\begin{aligned} A_{22}&\lesssim \Vert T^{\prime }_{\Delta _{j}\partial _{i}u}u_{2}^{i}\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim \sum \limits _{j-2\le j^{\prime }\le j} 2^{j}\Vert \Delta _{j^{\prime }}u_{2}^{i}\Vert _{L^{\infty }}\Vert \Delta _{j}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim \sum \limits _{j-2\le j^{\prime }\le j}2^{j^{\prime }(1-r_{2})}\Vert u_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\Vert \Delta _{j}u\Vert _{L^{2}}^{2}\nonumber \\&\lesssim 2^{-jr_{2}}\sum \limits _{j-2\le j^{\prime }\le j}2^{j^{\prime }}\Vert u_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\Vert \Delta _{j}u\Vert _{L^{2}}^{2}. \end{aligned}$$

(3.19)

Similar as (3.6) and (3.8) with few changes, we have

$$\begin{aligned} A_{23}\lesssim 2^{-jr_{2}}\Vert u_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{|j^{\prime }-j|\le 4}2^{j^{\prime }}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}},\end{aligned}$$

(3.20)

$$\begin{aligned} A_{24}\lesssim 2^{j}\Vert u_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j-3}2^{-j^{\prime }r_{2}}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}. \end{aligned}$$

(3.21)

Hence, combining (3.17)–(3.21), it follows that

$$\begin{aligned} A_{2}&\lesssim 2^{-jr_{2}}\Vert u_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}2^{j^{\prime }}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&+2^{j}\Vert u_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }} \sum \limits _{j^{\prime }\ge j}2^{-jr_{2}}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j}u\Vert _{L^{2}}. \end{aligned}$$

(3.22)

Similarly,

$$\begin{aligned} B_{2}&\lesssim 2^{-jr_{2}}\Vert b_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}2^{j^{\prime }}\Vert \Delta _{j^{\prime }}b\Vert _{L^{2}}\Vert \Delta _{j}b\Vert _{L^{2}}\nonumber \\&+\,2^{j}\Vert b_{2}\Vert _{B^{r_{2}}_{\infty ,\infty }} \sum \limits _{j^{\prime }\ge j}2^{-jr_{2}}\Vert \Delta _{j^{\prime }}b\Vert _{L^{2}}\Vert \Delta _{j}b\Vert _{L^{2}}. \end{aligned}$$

(3.23)

Inserting the estimates (3.11)–(3.16), (3.22) and (3.23) into (3.3) and (3.4), we have

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}^{2}+ca_{j}2^{2j\beta } \Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}^{2}\nonumber \\&\lesssim 2^{j(1-r_{1})}\Vert (u_{1},b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }} b)\Vert _{L^{2}}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}\nonumber \\&+\, 2^{j} \Vert (u_{1},b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j}2^{-j^{\prime }r_{1}}\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }} b)\Vert _{L^{2}}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}\nonumber \\&+\, 2^{-jr_{2}}\Vert (u_{2},b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\le j}2^{j^{\prime }}\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }} b)\Vert _{L^{2}}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}\nonumber \\&+\, 2^{j} \Vert (u_{2},b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }}\sum \limits _{j^{\prime }\ge j}2^{-j^{\prime }r_{2}}\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }} b)\Vert _{L^{2}}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}\nonumber \\&\equiv \, I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned}$$

(3.24)

Now, under the assumption ($a$), notice that the condition $r_{1}+r_{2}>1-2\beta$ implies that one of $r_{1}$ and $r_{2}$ must be bigger than $\frac{1-2\beta }{2}$, without loss of generality, we assume that $r_{1}>\frac{1-2\beta }{2}$. We first choose a real number $s$ such that

$$\begin{aligned} \frac{1-\beta -r_{1}}{\beta }<s<\min \left\{ \frac{r_{1}+\beta }{\beta },\frac{r_{2}+\beta }{\beta }\right\} . \end{aligned}$$

Multiplying (3.24) with $2^{-2j\beta s}$ and integrating with time variable, it follows that for $j\ge -1$,

$$\begin{aligned}&2^{-2js\beta }\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}+a_{j}2^{2j\beta (1-s)} \int \limits _{0}^{t}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}(\tau )\text{ d}\tau \nonumber \\&\lesssim 2^{-2js\beta }\int \limits _{0}^{t}(I_{1}+I_{2}+I_{3}+I_{4})\text{ d}\tau . \end{aligned}$$

(3.25)

Set

$$\begin{aligned} A(t):=\sup _{j\ge -1}2^{-js\beta }\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}, \quad B(t):=\sup _{j\ge -1} 2^{2j\beta (1-s)}\int \limits _{0}^{t}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}(\tau )\text{ d}\tau . \end{aligned}$$

Notice that $\frac{1-\beta -r_{1}}{\beta }<s<\min \left\{ \frac{r_{1}+\beta }{\beta },\frac{r_{2}+\beta }{\beta }\right\}$, by using Hölder’s inequality and Young inequality, we get

$$\begin{aligned} 2^{-2js\beta }\!\!\int \limits _{0}^{t}I_{1}\text{ d}\tau&\le \sum \limits _{j^{\prime }\le j} 2^{(j^{\prime }-j)(1-\beta -r_{1}-s\beta )}\!\!\int \limits _{0}^{t}\Vert (u_{1},b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }} (2^{-j^{\prime }s\beta }\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }}b)\Vert _{L^{2}})^{\frac{2\beta -1+r_{1}}{\beta }}\nonumber \\&\times (2^{j^{\prime }\beta (1-s)}\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }}b)\Vert _{L^{2}})^{\frac{1-\beta -r_{1}}{\beta }} (2^{j\beta (1-s)}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}})\text{ d}\tau \nonumber \\&\le C\left(\int \limits _{0}^{t}\Vert (u_{1},b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{1}}}A^{2}(\tau )\text{ d}\tau \right)^{\frac{2\beta -1+r_{1}}{2\beta }} \cdot (B(t))^{\frac{1-r_{1}}{2\beta }}\nonumber \\&\le C \int \limits _{0}^{t}\Vert (u_{1},b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{1}}}A^{2}(\tau )\text{ d}\tau +\varepsilon B(t). \end{aligned}$$

(3.26)

Similarly,

$$\begin{aligned} 2^{-2js\beta }\int \limits _{0}^{t}I_{2}\text{ d}\tau&\le \sum \limits _{j^{\prime }\ge j} 2^{(j^{\prime }-j)(s\beta -\beta -r_{1})}\int \limits _{0}^{t}\Vert (u_{1},b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }} (2^{-j^{\prime }\beta (1-s)}\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }}b)\Vert _{L^{2}})\nonumber \\&\times (2^{j^{\prime }\beta s}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}})^{\frac{2\beta -1+r_{1}}{\beta }} (2^{j\beta (1-s)}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}})^{\frac{1-\beta -r_{1}}{\beta }}\text{ d}\tau \nonumber \\&\le C\left(\int \limits _{0}^{t}\Vert (u_{1},b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{1}}}A^{2}(\tau )\text{ d}\tau \right)^{\frac{2\beta -1+r_{1}}{2\beta }} \cdot (B(t))^{\frac{1-r_{1}}{2\beta }}\nonumber \\&\le C \int \limits _{0}^{t}\Vert (u_{1},b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{1}}}A^{2}(\tau )\text{ d}\tau +\varepsilon B(t). \end{aligned}$$

(3.27)

For the last two terms, we have

$$\begin{aligned} 2^{-2js\beta }\int \limits _{0}^{t}I_{3}\text{ d}\tau&\le \sum \limits _{j^{\prime }\le j} 2^{(j^{\prime }-j)(s\beta +\beta +r_{2})}\int \limits _{0}^{t}\Vert (u_{2},b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }} (2^{-j^{\prime }s\beta }\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }}b)\Vert _{L^{2}})^{\frac{2\beta -1+r_{2}}{\beta }}\nonumber \\&\times (2^{j^{\prime }\beta (1- s)}\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }}b)\Vert _{L^{2}})^{\frac{1-\beta -r_{1}}{\beta }} (2^{j\beta (1-s)}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}})\text{ d}\tau \nonumber \\&\le C\left(\int \limits _{0}^{t}\Vert (u_{2},b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{2}}}A^{2}(\tau )\text{ d}\tau \right)^{\frac{2\beta -1+r_{2}}{2\beta }} \cdot (B(t))^{\frac{1-r_{2}}{2\beta }}\nonumber \\&\le C \int \limits _{0}^{t}\Vert (u_{2},b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{2}}}A^{2}(\tau )\text{ d}\tau +\varepsilon B(t);\end{aligned}$$

(3.28)

$$\begin{aligned} 2^{-2js\beta }\int \limits _{0}^{t}I_{4}\text{ d}\tau&\le \sum \limits _{j^{\prime }\ge j} 2^{(j^{\prime }-j)(s\beta -\beta -r_{2})}\int \limits _{0}^{t}\Vert (u_{2},b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }} (2^{j^{\prime }\beta (1-s)}\Vert (\Delta _{j^{\prime }}u,\Delta _{j^{\prime }}b)\Vert _{L^{2}})\nonumber \\&\times (2^{-js\beta }\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}})^{\frac{2\beta -1-r_{2}}{\beta }} (2^{j\beta (1-s)}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}})^{\frac{1-\beta -r_{2}}{\beta }}\text{ d}\tau \nonumber \\&\le C\left(\int \limits _{0}^{t}\Vert (u_{2},b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{2}}}A^{2}(\tau )\text{ d}\tau \right)^{\frac{2\beta -1+r_{2}}{2\beta }} \cdot (B(t))^{\frac{1-r_{2}}{2\beta }}\nonumber \\&\le C \int \limits _{0}^{t}\Vert (u_{2},b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{2}}}A^{2}(\tau )\text{ d}\tau +\varepsilon B(t). \end{aligned}$$

(3.29)

Now, notice that if $B(t)=2^{-2\beta (1-s)}\int _{0}^{t}\Vert (\Delta _{-1}u,\Delta _{-1}b)\Vert _{L^{2}}^{2}(\tau )\text{ d}\tau$, then we have

$$\begin{aligned} B(t)\le T2^{-2\beta }\cdot 2^{2\beta s}\Vert (\Delta _{-1}u,\Delta _{-1}b)\Vert _{L^{2}}^{2}\le T 2^{-\beta }A^{2}(t). \end{aligned}$$

(3.30)

Hence, by gathering (3.25)–(3.30) and choosing $\varepsilon$ sufficiently small, it follows that

$$\begin{aligned} A^{2}(t)\le C\int \limits _{0}^{t}(\Vert (\Delta _{j}u_{1},\Delta _{j}b_{1})\Vert _{B^{r_{1}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{1}}} +\Vert (\Delta _{j}u_{2},\Delta _{j}b_{2})\Vert _{B^{r_{2}}_{\infty ,\infty }}^{\frac{2\beta }{2\beta -1+r_{2}}})A^{2}(\tau )\text{ d}\tau . \end{aligned}$$

(3.31)

The estimate (3.31) and the Gronwall inequality yield that

$$\begin{aligned} A(t)=0, \text{ i.e.,} (u_{1},b_{1})\equiv (u_{2},b_{2}) \text{ a.e.} \text{ on} (x,t)\in \mathbb R ^{n}\times [0,T]. \end{aligned}$$

This completes the proof of Theorem 1.2 under the assumption ($a$).

For the case $(u_{1},b_{1})$ and $(u_{2},b_{2})$ satisfy the assumption ($b$), we will use the idea of the losing derivative estimate which was introduced by Chemin and Lermer [6].

Let $s\in (0,1)$. For $\lambda >0$, we set

$$\begin{aligned} A_{j}^{\lambda }(t)=2^{-js}e^{-\lambda \phi _{j}(t)}\Vert (\Delta _{j}u,\Delta _{j}b)\Vert _{L^{2}}, \end{aligned}$$

where $\phi _{j}(t)$ is defined by

$$\begin{aligned} \phi _{j}(t)=\int \limits _{0}^{t}\sum \limits _{j^{\prime }\le j+4}2^{j^{\prime }}(\Vert (\Delta _{j}u_{1},\Delta _{j}b_{1})(\tau )\Vert _{L^{\infty }}+\Vert (\Delta _{j}u_{2},\Delta _{j}b_{2}) (\tau )\Vert _{L^{\infty }})\text{ d}\tau . \end{aligned}$$

We get by (3.3) and (3.4) that

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}A_{j}^{\lambda }(t)+\lambda \phi ^{\prime }_{j}(t)A_{j}^{\lambda }(t)+a_{j}2^{2j\beta }A_{j}^{\lambda }(t)\nonumber \\&\lesssim 2^{-js}e^{-\lambda \phi _{j}(t)}(\Vert \Delta _{j}(u\cdot \nabla u_{1})\Vert _{L^{2}}+\Vert [u_{2},\Delta _{j}]\nabla -\sum \limits _{j^{\prime }>j}\partial _{i}\Delta _{j}u\Delta _{j^{\prime }}u_{2}^{i}\Vert _{L^{2}}\nonumber \\ \quad&+\,\Vert \Delta _{j}(b\cdot \nabla b_{1} )\Vert _{L^{2}}+\Vert \Delta _{j}(b_{2}\cdot \nabla b)\Vert _{L^{2}}+\Vert \Delta _{j}(u\cdot \nabla b_{1})\Vert _{L^{2}}\nonumber \\ \quad&+\,\Vert [u_{2},\Delta _{j}]\nabla u-\sum \limits _{j^{\prime }>j}\partial _{i}\Delta _{j}b\Delta _{j^{\prime }}u_{2}^{i}\Vert _{L^{2}}+\Vert \Delta _{j}(b\cdot \nabla u_{1})\Vert _{L^{2}}+\Vert \Delta _{j}(b_{2}\cdot \nabla u)\Vert _{L^{2}})(\tau )\text{ d}\tau \nonumber \\&\equiv \, J_{1}+J_{2}+\cdots +J_{8}, \end{aligned}$$

(3.32)

where we have used the facts $\langle \partial _{i}\Delta _{j}u\Delta _{j^{\prime }}u_{2}^{i},\Delta _{j}u\rangle =-\langle \Delta _{j^{\prime }}\partial _{i}u_{2}^{i}\Delta _{j}u,\Delta _{j}u\rangle =0$ and $\langle \partial _{i}\Delta _{j}b\Delta _{j^{\prime }}u_{2}^{i},\Delta _{j}b\rangle =-\langle \Delta _{j^{\prime }}\partial _{i}u_{2}^{i}\Delta _{j}b,\Delta _{j}b\rangle =0$.

Using the equality of Bony decomposition, we have

$$\begin{aligned} \Vert \Delta _{j}(u\cdot \nabla u_{1})\Vert _{L^{2}}\le \Vert \Delta _{j}(T_{u^{i}}\partial _{i} u_{1})\Vert _{L^{2}}+\Vert \Delta _{j}(T_{\partial _{i}u_{1}}u^{i})\Vert _{L^{2}}+\Vert \Delta _{j}R(u^{i},\partial _{i}u_{1})\Vert _{L^{2}}.\qquad \end{aligned}$$

(3.33)

By the equality $\Delta _{j}(T_{u^{i}}\partial _{i}u_{1})=\sum _{|j^{\prime }-j|\le 4}\Delta _{j}(S_{j^{\prime }-1}u^{i}\partial _{i}\Delta _{j^{\prime }}u_{1})$ and Bernstein’s Lemma, we get

$$\begin{aligned} \Vert \Delta _{j}(T_{u^{i}}\partial _{i} u_{1})(\tau )\Vert _{L^{2}}\lesssim&\sum \limits _{|j^{\prime }-j|\le 4}2^{j^{\prime }}\sum \limits _{j^{\prime \prime }\le j^{\prime }-2}\Vert \Delta _{j^{\prime \prime }}u\Vert _{L^{2}}\Vert \Delta _{j^{\prime }}u_{1}\Vert _{L^{\infty }}\nonumber \\ \lesssim&\sum \limits _{|j^{\prime }-j|\le 4}2^{j^{\prime }}\sum \limits _{j^{\prime \prime }\le j^{\prime }-2} 2^{j^{\prime \prime }s}e^{\lambda \phi _{j^{\prime \prime }}(\tau )}A_{j^{\prime \prime }}^{\lambda }\Vert \Delta _{j^{\prime }}u_{1}\Vert _{L^{\infty }}\nonumber \\ \lesssim&\sum \limits _{j^{\prime }\le j+2} 2^{j^{\prime }s} e^{\lambda \phi _{j^{\prime }}(\tau )}A_{j^{\prime }}^{\lambda }(\tau )\phi _{j}^{\prime }(\tau ). \end{aligned}$$

(3.34)

By the equalities (3.7) and (3.9), we can estimate the last two terms of (3.33) as

$$\begin{aligned} \Vert \Delta _{j}(T_{\partial _{i}u_{1}}u^{i})(\tau )\Vert _{L^{2}}&\lesssim \sum \limits _{|j^{\prime }-j|\le 4}\sum \limits _{j^{\prime \prime }\le j^{\prime }-2}2^{j^{\prime \prime }}\Vert \Delta _{j^{\prime \prime }}u_{1}\Vert _{L^{\infty }}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\nonumber \\&\lesssim \sum \limits _{|j^{\prime }-j|\le 4} 2^{j^{\prime }s}e^{\lambda \phi _{j^{\prime }}(\tau )}A_{j^{\prime }}^{\lambda }(\tau )\sum \limits _{j^{\prime \prime }\le j^{\prime }-2}2^{j^{\prime \prime }}\Vert \Delta _{j^{\prime \prime }}u_{1}\Vert _{L^{\infty }}\nonumber \\&\lesssim \sum \limits _{j^{\prime }\le j+2} 2^{j^{\prime }s} e^{\lambda \phi _{j^{\prime }}(\tau )}A_{j^{\prime }}^{\lambda }(\tau )\phi _{j}^{\prime }(\tau );\end{aligned}$$

(3.35)

$$\begin{aligned} \Vert \Delta _{j}R(u^{i},\partial _{i}u_{1})(\tau )\Vert _{L^{2}}&\lesssim \sum \limits _{j^{\prime },j^{\prime \prime }\ge j-3;|j^{\prime }-j^{\prime \prime }|\le 1} 2^{j}\Vert \Delta _{j^{\prime }}u\Vert _{L^{2}}\Vert \Delta _{j^{\prime \prime }}u_{1}\Vert _{L^{\infty }}\nonumber \\&\lesssim \sum \limits _{j^{\prime },j^{\prime \prime }\ge j-3;|j^{\prime }-j^{\prime \prime }|\le 1} 2^{j} 2^{j^{\prime }s}e^{\lambda \phi _{j^{\prime }}(\tau )}A_{j^{\prime }}^{\lambda }(\tau )\Vert \Delta _{j^{\prime \prime }}u_{1}\Vert _{L^{\infty }}\nonumber \\&\lesssim 2^{j}\sum \limits _{j^{\prime }\ge j-3} 2^{j^{\prime }(s-1)}e^{\lambda \phi _{j^{\prime }}(\tau )}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau ). \end{aligned}$$

(3.36)

Combining (3.33)–(3.36) together, we can estimate $J_{1}$ as

$$\begin{aligned} \int \limits _{0}^{t}J_{1}(\tau )\text{ d}\tau&= \int \limits _{0}^{t}2^{-js}e^{-\lambda \phi _{j}(\tau )} \Vert \Delta _{j}(u\cdot \nabla u_{1})(\tau )\Vert _{L^{2}}\text{ d}\tau \nonumber \\&\lesssim \sum \limits _{j^{\prime }\le j}2^{(j^{\prime }-j)s}\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j}(\tau )\text{ d}\tau \nonumber \\ \quad&+\,\sum \limits _{j^{\prime }\ge j} 2^{-(j^{\prime }-j)(1-s)}\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau . \end{aligned}$$

(3.37)

Similarly,

$$\begin{aligned} \int \limits _{0}^{t}(J_{3}+J_{4}+J_{5}&+ J_{7}+J_{8})(\tau )\text{ d}\tau \lesssim \sum \limits _{j^{\prime }\le j}2^{(j^{\prime }-j)s}\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j}(\tau )\text{ d}\tau \nonumber \\ \quad&+ \sum \limits _{j^{\prime }\ge j} 2^{-(j^{\prime }-j)(1-s)}\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau . \end{aligned}$$

(3.38)

To estimate the term $J_{2}$, we first notice that there holds

$$\begin{aligned} \Vert [u_{2},\Delta _{j}]\nabla \!-\!\!\sum \limits _{j^{\prime }>j}\partial _{i}\Delta _{j}u\Delta _{j^{\prime }}u_{2}^{i}\Vert _{L^{2}}\, \le \, \Vert [T_{u_{2}^{i}},\Delta _{j}]\partial _{i}u\Vert _{L^{2}}\!+\! \Vert T^{\prime }_{\Delta _{j}\partial _{i}u}u_{2}^{i}\!-\!\!\sum \limits _{j^{\prime }>j}\partial _{i}\Delta _{j}u\Delta _{j^{\prime }}u_{2}^{i}\Vert _{L^{2}}\nonumber \\&+\Vert \Delta _{j}(T_{\partial _{i}u}u_{2}^{i})\Vert _{L^{2}}+\Vert \Delta _{j}R(u_{2}^{i},\partial _{i}u)\Vert _{L^{2}}. \end{aligned}$$

(3.39)

Similar to derivation of estimate (3.18), we have

$$\begin{aligned} \Vert [T_{u_{2}^{i}},\Delta _{j}]\partial _{i}u(\tau )\Vert _{L^{2}}&\lesssim \sum \limits _{|j^{\prime }-j|\le 4}\Vert \nabla S_{j^{\prime }-1}u_{2}\Vert _{L^{\infty }}\Vert \Delta _{j}u\Vert _{L^{2}}\nonumber \\&\lesssim \sum \limits _{|j^{\prime }-j|\le 4} 2^{(j^{\prime }-1)}\Vert u_{2}\Vert _{L^{\infty }}2^{j^{\prime }s}e^{\lambda \phi _{j^{\prime }}(\tau )}A_{j^{\prime }}^{\lambda }(\tau )\nonumber \\&\lesssim \sum \limits _{|j^{\prime }-j|\le 4} 2^{j^{\prime }s} e^{\lambda \phi _{j^{\prime }}(\tau )}A_{j^{\prime }}^{\lambda }(\tau ) \phi ^{\prime }_{j}(\tau ). \end{aligned}$$

(3.40)

Notice that

$$\begin{aligned} T^{\prime }_{\Delta _{j}\partial _{i}u}u_{2}^{i}-\sum \limits _{j^{\prime }>j}\partial _{i}\Delta _{j}u\Delta _{j^{\prime }}u_{2}^{i} =\sum \limits _{j-2\le j^{\prime }\le j} S_{j^{\prime }+2}\Delta _{j}\partial _{i}u \Delta _{j^{\prime }}u_{2}, \end{aligned}$$

it gives by Bernstein’s Lemma 2.2 that

$$\begin{aligned} \Vert (T^{\prime }_{\Delta _{j}\partial _{i}u}u_{2}^{i}-\sum \limits _{j^{\prime }>j}\partial _{i}\Delta _{j}u\Delta _{j^{\prime }}u_{2}^{i})(\tau )\Vert _{L^{2}}&\lesssim \sum \limits _{j-2\le j^{\prime }\le j}\Vert S_{j^{\prime }+2}\Delta _{j}\partial _{i}u\Vert _{L^{2}}\Vert \Delta _{j^{\prime }}u_{2}\Vert _{L^{\infty }}\nonumber \\&\lesssim \sum \limits _{j-2\le j^{\prime }\le j}2^{j^{\prime }}\Vert \Delta _{j}u\Vert _{L^{2}}\Vert \Delta _{j^{\prime }}u_{2}\Vert _{L^{\infty }}\nonumber \\&\lesssim 2^{js}e^{\lambda \phi _{j}(\tau )}A_{j}^{\lambda }(\tau )\phi ^{\prime }_{j}(\tau ). \end{aligned}$$

(3.41)

The last two terms can be estimate similar to (3.8) and (3.10),

$$\begin{aligned} \Vert \Delta _{j}(T_{\partial _{i}u}u_{2}^{i})(\tau )\Vert _{L^{2}}&\lesssim \sum \limits _{j^{\prime }\le j+2}2^{j^{\prime }s}e^{\lambda \phi _{j^{\prime }}(\tau )} A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j}(\tau );\end{aligned}$$

(3.42)

$$\begin{aligned} \Vert \Delta _{j}R(u_{2}^{i},\partial _{i}u)(\tau )\Vert _{L^{2}}&\lesssim 2^{j}\sum \limits _{j^{\prime }\ge j-3}2^{j^{\prime }(s-1)}e^{\lambda \phi _{j^{\prime }}(\tau )}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau ). \end{aligned}$$

(3.43)

We can estimate the term $\Vert [u_{2}\Delta _{j}]\nabla u-\sum \limits _{j^{\prime }>j}\partial _{i}\Delta _{j}b\Delta _{j^{\prime }}u_{2}^{i}\Vert _{L^{2}}$ in the same way. Hence, combining (3.39)–(3.43), we get

$$\begin{aligned} \int \limits _{0}^{t}(J_{2}+J_{6})(\tau )\text{ d}\tau&\lesssim \sum \limits _{j^{\prime }\le j}2^{(j^{\prime }-j)s}\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau \nonumber \\&+\sum \limits _{j^{\prime }\ge j} 2^{-(j^{\prime }-j)(1-s)}\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau . \end{aligned}$$

(3.44)

Inserting (3.37), (3.38) and (3.44) into (3.32), it follows that

$$\begin{aligned}&A_{j}^{\lambda }(t)+\lambda \int \limits _{0}^{t}\phi ^{\prime }_{j}(\tau )A_{j}^{\lambda }(\tau )\text{ d}\tau +a_{j}2^{2j\beta }\int \limits _{0}^{t}A_{j}^{\lambda }(\tau )\text{ d}\tau \nonumber \\&\lesssim \sum \limits _{j^{\prime }\le j}2^{(j^{\prime }-j)s}\!\!\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda } (\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau \nonumber \\&+\!\sum \limits _{j^{\prime }\ge j} 2^{-(j^{\prime }-j)(1-s)}\!\!\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau . \end{aligned}$$

(3.45)

Notice that $\phi ^{\prime }_{j}(\tau )=\phi ^{\prime }_{j^{\prime }}(\tau )+(\phi ^{\prime }_{j}(\tau )-\phi ^{\prime }_{j^{\prime }}(\tau ))$ and $\phi ^{\prime }_{j}(\tau )-\phi ^{\prime }_{j^{\prime }}(\tau )\ge 0$ for $j\ge j^{\prime }$ imply that

$$\begin{aligned} \int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}(\phi ^{\prime }_{j}(\tau )-\phi ^{\prime }_{j^{\prime }}(\tau ))\text{ d}\tau \lesssim \frac{1}{\lambda } \quad \text{ for} j\ge j^{\prime }. \end{aligned}$$

Hence

$$\begin{aligned}&\sum \limits _{j^{\prime }\le j}2^{(j^{\prime }-j)s}\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau \nonumber \\&\lesssim \sum \limits _{j^{\prime }\le j} 2^{(j^{\prime }-j)s}\int \limits _{0}^{t}A_{j}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau +\frac{1}{\lambda }\sum \limits _{j^{\prime }\le j}2^{(j^{\prime }-j)s}\sup _{\tau \in [0,t]}A_{j^{\prime }}^{\lambda }(\tau )\nonumber \\&\lesssim \sup _{j\ge -1}\int \limits _{0}^{t}A_{j^{\prime }}^{\lambda }\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau +\frac{1}{\lambda }\sum \limits _{j^{\prime }\le j}2^{(j^{\prime }-j)s}\sup _{\tau \in [0,t]}A_{j^{\prime }}^{\lambda }(\tau ). \end{aligned}$$

(3.46)

Notice that $\phi _{j^{\prime }}(t)-\phi _{j}(t)$ is an increasing function with $t$ for $j^{\prime }\ge j$, and $\phi _{j^{\prime }}(t)-\phi _{j}(t)\le (j^{\prime }-j)(\Vert (u_{1},b_{1})\Vert _{L^{1}(0,t;B^{1}_{\infty ,\infty })}+\Vert (u_{2},b_{2})\Vert _{L^{1}(0,t;B^{1}_{\infty ,\infty })})$, we have

$$\begin{aligned}&\sum \limits _{j^{\prime }\ge j} 2^{-(j^{\prime }-j)(1-s)}\int \limits _{0}^{t}e^{\lambda (\phi _{j^{\prime }}(\tau )-\phi _{j}(\tau ))}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau \nonumber \\&\lesssim \sum \limits _{j^{\prime }\ge j}2^{-(j^{\prime }-j)(1-s)}e^{\lambda (\phi _{j^{\prime }}(t))-\phi _{j}(t)}\int \limits _{0}^{t}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau \nonumber \\&\lesssim \sum \limits _{j^{\prime }\ge j}2^{-(j^{\prime }-j)(1-s)}e^{\lambda (j^{\prime }-j)(\Vert (u_{1},b_{1})\Vert _{L^{1}(0,t;B^{1}_{\infty ,\infty })}+\Vert (u_{2},b_{2})\Vert _{L^{1}(0,t;B^{1}_{\infty ,\infty })})} \int \limits _{0}^{t}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau \nonumber \\&\lesssim \sum \limits _{j^{\prime }\ge j}2^{-(j^{\prime }-j)(1-s)}\int \limits _{0}^{t}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau \nonumber \\&\lesssim \sup _{j^{\prime }\ge -1}\int \limits _{0}^{t}A_{j^{\prime }}^{\lambda }(\tau )\phi ^{\prime }_{j^{\prime }}(\tau )\text{ d}\tau , \end{aligned}$$

(3.47)

where we have used the assumption

$$\begin{aligned} \lambda (\Vert (u_{1},b_{1})\Vert _{L^{1}(0,t;B^{1}_{\infty ,\infty })}+\Vert (u_{2},b_{2})\Vert _{L^{1}(0,t;B^{1}_{\infty ,\infty })}) \le (1-s)\text{ ln}2. \end{aligned}$$

(3.48)

Summing up (3.45)–(3.47) together, it follows that

$$\begin{aligned}&\sup _{j\ge -1;\tau \in [0,t]}A_{j}^{\lambda }(\tau )+\lambda \sup _{j\ge -1}\int \limits _{0}^{t}A_{j}^{\lambda }(\tau )\phi ^{\prime }_{j}(\tau )\text{ d}\tau +\sup _{j\ge -1}a_{j}2^{2\beta j}\int \limits _{0}^{t}A_{j}^{\lambda }(\tau )\text{ d}\tau \nonumber \\&\le C\sup _{j\ge -1}\int \limits _{0}^{t}A_{j}^{\lambda }(\tau )\text{ d}\tau +\frac{C}{\lambda }\sup _{j\ge -1;\tau \in [0,t]}A_{j}^{\lambda }(\tau ). \end{aligned}$$

Taking $\lambda$ big enough and using the Gronwall inequality yield that

$$\begin{aligned} \sup _{j\ge -1;\tau \in [0,t]}A_{j}^{\lambda }(\tau )=0. \end{aligned}$$

Hence, $(u_{1},b_{1})= (u_{2},b_{2})$ a.e. $\mathbb R ^{n}\times [0,t]$. On the other hand, under the assumption ($b$), we can choose the $t>0$ small enough such that (3.48) holds, then by using the standard continuity argument, we obtain $(u_{1},b_{1})=(u_{2},b_{2})$ on $\mathbb R ^{n}\times [0,T]$, and the proof of Theorem 1.2 is complete. $\Box$

4 Proof of Theorem 1.3

In this section, we give the proof of Theorem 1.3. We first introduce a lemma of inhomogeneous Sobolev inequality identified due to P. Gerard, Y. Meyer and F. Oru [9] (see also [13], for another demonstration of these inequalities):

Lemma 4.1

Let $0<\alpha <\gamma$, $1<p<\infty$ and $\frac{p}{q}=(1-\frac{\alpha }{\gamma })$. Then, we have

$$\begin{aligned} \Vert f\Vert _{L^{q}}\lesssim \Vert f\Vert _{H^{\alpha }_{p}}^{1-\frac{\alpha }{\gamma }}\Vert f\Vert _{B^{\alpha -\gamma }_{\infty ,\infty }}^{\frac{\alpha }{\gamma }} \end{aligned}$$

for all $f\in H^{\alpha }_{p}\cap B^{\alpha -\gamma }_{\infty ,\infty }$. Here $H^{\alpha }_{p}:=\{f:f\in \mathcal S ^{\prime }(\mathbb R ^{n}),\Vert f\Vert _{H^{\alpha }_{p}}= \Vert \mathcal F ^{-1}\{(1+|\xi |^{2})^{\frac{\alpha }{2}}\mathcal F f\}\Vert _{L^{p}}<\infty \}$.

Notice that whether $i=1$ or $2$, the classical interpolation inequality implies that the weak solution $(u_{i},b_{i})\in L^{\infty }_{T}L^{2}\cap L^{2}_{T} H^{\beta }$ still belongs to $L^{\frac{2\beta }{r_{i}}}_{T}H^{r_{i}}$. Thus, under the assumption of Theorem 1.3, we get that $(u_{i},b_{i})$ belongs to $L^{\frac{2\beta }{r_{i}}}_{T}H^{r_{1}}\cap L^{\frac{2\beta }{2\beta -1-r_{i}}}(0,T;B^{-r_{i}}_{\infty ,\infty })$, then the above Lemma 4.1 and the classical interpolation inequality imply that $(u_{i},b_{i})\in L^{\frac{4\beta }{2\beta -1}}(0,T;L^{4}(\mathbb R ^{n}))$. Hence, when $3\le n\le 4(2\beta -1)$ (here, we need $\beta \ge \frac{7}{8}$), it follows that Theorem 1.3 will be a straightforward corollary of the following theorem:

Theorem 4.2

Let $\beta \in [\frac{7}{8},1]$, $3\le n\le 4(2\beta -1)$, $T>0$ and $(u_{0},b_{0})\in L^{2}(\mathbb R ^{n})$ with $\mathrm div u_{0}=\mathrm div b_{0}=0$. Let $(u_{1},b_{1})$ and $(u_{2},b_{2})$ be two weak solutions of the GMHD system (1.1)–(1.3) with the same initial value $(u_{0},b_{0})$ satisfying

$$\begin{aligned}&(u_{1},b_{1})\in \L ^{q}(0,T; L^{p})\cap L^{\frac{2\beta }{2\beta -1-r_{1}}}(0,T;B^{-r_{1}}_{\infty ,\infty }) \end{aligned}$$

and

$$\begin{aligned}&(u_{2},b_{2})\in \L ^{q}(0,T; L^{p})\cap L^{\frac{2\beta }{2\beta -1-r_{2}}}(0,T; B^{-r_{2}}_{\infty ,\infty }) \end{aligned}$$

with $0< r_{1},r_{2}\le 2\beta -1$, and for some $p\in [\frac{n}{2\beta -1},\infty )$ and $q\in (\frac{2\beta }{2\beta -1},\infty ]$. Then, $(u_{1},b_{1})\equiv (u_{2},b_{2})$ a.e. on $\mathbb R ^{n}\times (0,T)$.

In order to prove Theorem 4.2, we need the following two lemmas.

Lemma 4.3

Suppose $(u_{0},b_{0})\in L^{s}(\mathbb R ^{n}), s\ge \frac{n}{2\beta -1}$. Then, there exit $T_{0}>0$ and a unique solution $(u,b)\in BC([0,T_{0});L^{s})$ of the GMHD system (1.1)–(1.4) such that

$$\begin{aligned} \sup _{0<t<T_{0}}t^{\frac{2\beta -1}{2\beta }}\Vert (u(\cdot ,t),b(\cdot ,t))\Vert _{L^{\infty }}<\infty \text{ and} \lim _{t\rightarrow 0}t^{\frac{2\beta -1}{2\beta }}\Vert (u(\cdot ,t),b(\cdot ,t))\Vert _{L^{\infty }}=0. \end{aligned}$$

Moreover, $(u,b)$ is smooth on $(0,T_{0})\times \mathbb R ^{n}$; more precisely, $(u,b)\in C^{\infty }((0,T_{0});\widetilde{B}^{\sigma }_{\infty ,\infty })$ for all $\sigma >0$. Furthermore, if we denote by $(0,T_{*})$ be the maximal interval such that $(u,b)$ solves the system (1.1)–(1.4), then for $r\le 2\beta -1$, there holds

$$\begin{aligned} \lim _{\tau \rightarrow T_{*}}(T_{*}-\tau )^{\frac{2\beta -1-r}{2\beta }}\Vert (u(\cdot ,\tau ),b(\cdot ,\tau ))\Vert _{B^{-r}_{\infty ,\infty }}\ge C \end{aligned}$$

with constant $C>0$ independent of $T_{*}$ and $r$. Here, $BC$ denotes the class of bounded and continuous functions, and $\widetilde{B}^{\sigma }_{\infty ,\infty }:=\overline{\mathcal{S }(\mathbb R ^{n})}^{B^{\sigma }_{\infty ,\infty }}$.

Proof

The proof of this lemma is similar to that of Giga [12] (see also [14, 16, 19, 20]).$\square$

Lemma 4.4

Let $\beta \in [\frac{7}{8},1]$, $3\le n\le 4(2\beta -1)$, $T>0$ and $(u_{0},b_{0})\in L^{2}(\mathbb R ^{n})$ with $\mathrm div u_{0}=\mathrm div b_{0}=0$. Let $(u,b)$ be a weak solution of the GMHD (1.1)–(1.4) satisfying

$$\begin{aligned}&(u,b)\in \L ^{q}(0,T; L^{p})\cap L^{\frac{2\beta }{2\beta -1-r}}(0,T;B^{-r}_{\infty ,\infty }) \end{aligned}$$

(4.1)

with $0< r\le 2\beta -1$, $p\ge \frac{n}{2\beta -1}$ and $q> \frac{2\beta }{2\beta -1}$. Then, we have

$$\begin{aligned} \sup _{0<t<T}t^{\frac{2\beta -1}{2\beta }}\Vert (u,b)(\cdot ,t)\Vert _{L^{\infty }}<\infty \text{ and} \lim _{t\rightarrow 0}t^{\frac{2\beta -1}{2\beta }}\Vert (u,b)(\cdot ,t)\Vert _{L^{\infty }}=0. \end{aligned}$$

Proof

If we consider $t_{0}\in (0,T)$, there exists $\tau _{0}\in (0,t_{0})$ such that $u(\cdot ,\tau _{0})\in L^{p}(\mathbb R ^{n})$. By Lemma 4.3, we can establish a local-in-time solution $(\widetilde{u},\widetilde{b})\in C([\tau _{0},\tau );L^{p}(\mathbb R ^{n}))$ with $\widetilde{u}(\cdot ,\tau _{0})=u(\cdot ,\tau _{0})$ and $\widetilde{b}(\cdot ,\tau _{0})=b(\cdot ,\tau _{0})$. Let $\tau ^{*}$ be the supremum of the $\tau$ such that we have a solution in $C([\tau _{0},\tau ^{*});L^{p}(\mathbb R ^{n}))$. Moreover, we have $(\widetilde{u},\widetilde{b})\equiv (u,b)$ on $[\tau _{0},\min \{\tau ^{*},T\})$. But then $(\widetilde{u},\widetilde{b})\in C([\tau _{0},\min \{\tau ^{*},T\});L^{p}(\mathbb R ^{n}))$, the condition (4.1) and the last assertion of Lemma 4.3 give that $(\widetilde{u},\widetilde{b})$ can be extended beyond $\min \{\tau ^{*},T\}$. So that $\tau ^{*}\ge T$, thus $(u,b)\in C([\tau _{0},T);L^{p}(\mathbb R ^{n}))$ and satisfies $\sup _{\tau _{0}<t<T}(t-\tau _{0})^{\frac{2\beta -1}{2\beta }}\Vert (u,b)\Vert _{L^{\infty }}<\infty$. To complete the proof, it remains to prove that $\sup _{0<t<\tau _{0}} t^{\frac{2\beta -1}{2\beta }}\Vert (u,b)\Vert _{L^{\infty }}<\infty$ and $\lim _{t\rightarrow 0}t^{\frac{2\beta -1}{2\beta }}\Vert (u,b)(\cdot ,t)\Vert _{L^{\infty }}=0$. We use an argument similar to that of R. May [20] to prove this assertion.

Let $(t_{n})_{n}\in (0,\frac{T}{2})$ be a sequence which tends to $0$. Considering the functions $(u_{n},b_{n})_{n}$ be defined on $[0,\frac{T}{2})$ by $u_{n}(t):=u(t+t_{n})$ and $b_{n}(t):=b(t+t_{n})$. It is sufficient to prove that $\sup _{0<t<\delta }t^{\frac{2\beta -1}{2\beta }}\Vert (u_{n}(t),b_{n}(t))\Vert _{L^{\infty }}$ tend to $0$ uniformly with respect to $n$ when $\delta$ tends to $0$.

For $\mu \in \mathbb R$ and $\delta >0$, define

$$\begin{aligned} h_{n}(\mu ,\delta )&:= \sup _{0<t<\delta }t^{\frac{\mu +2\beta -1}{2\beta }}\Vert (u_{n}(t),b_{n}(t))\Vert _{B^{\mu }_{\infty ,\infty }};\\ \Theta (\delta )&:= \sup _{0<t_{0}<\frac{T}{2}}\Vert (u,b)\Vert _{L^{\frac{2\beta }{2\beta -1-r}}((t_{0},t_{0}+\delta );B^{-r}_{\infty ,\infty })}. \end{aligned}$$

Let $\sigma \in (r,2\beta -1)$ and $\delta _{0}\in (0,\frac{T}{2})$ to be chosen later, let $n\in \mathbb N ,\delta \in (0,\delta _{0}]$ and $t\in (0,\delta ]$. Denote $a=a(n,t)$ belonging to the interval $[\frac{t}{4},\frac{t}{2}]$ such that

$$\begin{aligned} \Vert (u_{n}(a),b_{n}(a))\Vert _{B^{-r}_{\infty ,\infty }}=\inf _{\tau \in [\frac{t}{4},\frac{t}{2}]}\Vert (u_{n}(\tau ),b_{n}(\tau ))\Vert _{B^{-r}_{\infty ,\infty }}. \end{aligned}$$

Notice that (see Lemma 2.6)

$$\begin{aligned} u_{n}(t)&= e^{-(t-a)(-\Delta )^{\beta }}u_{n}(a)-\int \limits _{a}^{t}e^{-(t-\tau )(-\Delta )^{\beta }} \mathbb P \nabla \cdot (u_{n}\otimes u_{n}+b_{n}\otimes b_{n})\text{ d}\tau \nonumber \\&:= I_{n}(t)+J_{n}(t);\end{aligned}$$

(4.2)

$$\begin{aligned} b_{n}(t)&= e^{-(t-a)(-\Delta )^{\beta }}b_{n}(a)-2\int \limits _{a}^{t}e^{-(t-\tau )(-\Delta )^{\beta }} \mathbb P \nabla \cdot (u_{n}\otimes b_{n})\text{ d}\tau \nonumber \\&:= \widetilde{I}_{n}(t)+\widetilde{J}_{n}(t). \end{aligned}$$

(4.3)

We will estimate the $B^{\sigma }_{\infty ,\infty }$ norm of $I_{n}(t),\widetilde{I}_{n}(t),J_{n}(t)$ and $\widetilde{J}_{n}(t)$. According to the second assertion of the Proposition 2.4 and the definition of $a=a(n,t)$, we have

$$\begin{aligned} \Vert I_{n}(t)\Vert _{B^{\sigma }_{\infty ,\infty }}&\lesssim (t-a)^{-\frac{r+\sigma }{2\beta }}\Vert u_{n}(a)\Vert _{B^{-r}_{\infty ,\infty }}\nonumber \\&\lesssim t^{-\frac{2\beta -1}{2\beta }}\Vert u_{n}\Vert _{L^{\frac{2\beta }{2\beta -1-r}}([\frac{t}{4},\frac{t}{2}];B^{-r}_{\infty ,\infty })}\nonumber \\&\lesssim t^{-\frac{2\beta -1+\sigma }{2\beta }}\Theta (\delta ). \end{aligned}$$

(4.4)

Similarly, $\Vert \widetilde{I}_{n}(t)\Vert _{B^{\sigma }_{\infty ,\infty }}\lesssim t^{-\frac{2\beta -1-r}{2\beta }}\Theta (\delta )$. On the other hand, notice that

$$\begin{aligned} J_{n}(t)&= \int \limits _{a}^{t}e^{-(t-\tau )(-\Delta )^{\beta }} \mathbb P \nabla \cdot (u_{n}\otimes u_{n}+b_{n}\otimes b_{n})\text{ d}\tau \\&:= \mathbb I _{Oss}(1_{[a,t)}(u_{n}\otimes u_{n}))+\mathbb I _{Oss}(1_{[a,t)}(b_{n}\otimes b_{n})). \end{aligned}$$

By using Propositions 2.4 and 2.7, it follows that the operator $\mathbb I _{Oss}$ is continuous from $\widetilde{L}^{\frac{2\beta }{2\beta -1-r}}_{T}(B^{\sigma -r}_{\infty ,\infty })$ to $L^{\infty }_{T}(B^{\sigma }_{\infty ,\infty })$. Hence, we deduce easily that

$$\begin{aligned} \Vert J_{n}(t)\Vert _{B^{\sigma }_{\infty ,\infty }}&\lesssim \Vert u_{n}\otimes u_{n}\Vert _{\widetilde{L}_{T}^{\frac{2\beta }{2\beta -1-r}}B^{\sigma -r}_{\infty ,\infty }} +\Vert b_{n}\otimes b_{n}\Vert _{\widetilde{L}_{T}^{\frac{2\beta }{2\beta -1-r}}B^{\sigma -r}_{\infty ,\infty }}\nonumber \\&\lesssim \Vert u_{n}\Vert _{\widetilde{L}^{\frac{2\beta }{2\beta -1-r}}([a,t);B^{-r}_{\infty ,\infty })} \Vert u_{n}\Vert _{L^{\infty }([a,t);B^{\sigma }_{\infty ,\infty })}\nonumber \\&+\Vert b_{n}\Vert _{\widetilde{L}^{\frac{2\beta }{2\beta -1-r}}([a,t);B^{-r}_{\infty ,\infty })} \Vert b_{n}\Vert _{L^{\infty }([a,t);B^{\sigma }_{\infty ,\infty })}\nonumber \\&\lesssim \Vert (u_{n},b_{n})\Vert _{L^{\frac{2\beta }{2\beta -1-r}}([a,t); B^{-r}_{\infty ,\infty })}\sup _{a\le \tau <t}\Vert (u_{n}(\tau ),b_{n}(\tau ))\Vert _{B^{\sigma }_{\infty ,\infty }}\nonumber \\&\lesssim t^{-\frac{\sigma +2\beta -1}{2\beta }}\Theta (\delta )h_{n}(\sigma ,\delta )\nonumber \\&\lesssim t^{-\frac{\sigma +2\beta -1}{2\beta }}\Theta (\delta _{0})h_{n}(\sigma ,\delta ), \end{aligned}$$

(4.5)

where we have used the Proposition 2.8 in the second inequality above. The estimate of $\widetilde{J}_{n}(t)$ can be derived in a similar way, that is, $\Vert \widetilde{J}_{n}(t)\Vert _{B^{\sigma }_{\infty ,\infty }}\lesssim t^{-\frac{\sigma +2\beta -1}{2\beta }}\Theta (\delta _{0})h_{n}(\sigma ,\delta )$. The inequalities (4.2)–(4.5) imply that there exists a constant $C_{1}>0$ independent of $t$, $\delta$ and $n$ such that

$$\begin{aligned} h_{n}(\sigma ,\delta )\le C_{1}\Theta (\delta )+C_{1}\Theta (\delta _{0})h_{n}(\sigma ,\delta ). \end{aligned}$$

By choosing $\delta _{0}$ small enough so that $\Theta (\delta _{0})$ is less than $\frac{1}{2C_{1}}$ (which is possible since $\Theta (\delta _{0})\rightarrow 0$ as $\delta _{0}\rightarrow 0$). Hence, the previous inequality gives that

$$\begin{aligned} h_{n}(\sigma ,\delta )\le 2 C_{1}\Theta (\delta _{0}). \end{aligned}$$

(4.6)

We return now to Eqs. (4.2) and (4.3), this time to estimate the $B^{-r}_{\infty ,\infty }$ norm of $I_{n}(t)$, $\widetilde{I}_{n}(t)$, $J_{n}(t)$ and $\widetilde{J}_{n}(t)$. Again, according to the second assertion of Proposition 2.4 and the definition of $a(n,t)$, it follows that

$$\begin{aligned} \Vert I_{n}(t)\Vert _{B^{-r}_{\infty ,\infty }}\lesssim \Vert u_{n}(a)\Vert _{B^{-r}_{\infty ,\infty }}\lesssim t^{-\frac{2\beta -1-r}{2\beta }}\Theta (\delta );\end{aligned}$$

(4.7)

$$\begin{aligned} \Vert \widetilde{I}_{n}(t)\Vert _{B^{-r}_{\infty ,\infty }}\lesssim \Vert b_{n}(a)\Vert _{B^{-r}_{\infty ,\infty }}\lesssim t^{-\frac{2\beta -1-r}{2\beta }}\Theta (\delta ). \end{aligned}$$

(4.8)

On the other hand, from Proposition 2.4 and the action of the pseudo-differential operator $\mathbb P \nabla$ on Besov space (see the second assertion of Proposition 2.7), we can deduce the following estimates

$$\begin{aligned} \Vert J_{n}(t)\Vert _{B^{-r}_{\infty ,\infty }}&\lesssim \int \limits _{a}^{t}(t-\tau )^{-\frac{1-\sigma }{2\beta }} (\Vert \mathbb P \nabla \cdot (u_{n}\otimes u_{n})\Vert _{B^{\sigma -r-1}_{\infty ,\infty }}+\Vert \mathbb P \nabla \cdot (b_{n}\otimes b_{n})\Vert _{B^{\sigma -r-1}_{\infty ,\infty }}(\tau )\text{ d}\tau \nonumber \\&\lesssim t^{\frac{\sigma +2\beta -1}{2\beta }}(\sup _{\frac{t}{4}<\tau <t}\Vert u_{n}(\tau )\Vert _{B^{-r}_{\infty ,\infty }}\sup _{\frac{t}{4}<\tau <t}\Vert u_{n}(\tau )\Vert _{B^{\sigma }_{\infty ,\infty }}\nonumber \\&+\sup _{\frac{t}{4}<\tau <t}\Vert b_{n}(\tau )\Vert _{B^{-r}_{\infty ,\infty }} \sup _{\frac{t}{4}<\tau <t}\Vert b_{n}(\tau )\Vert _{B^{\sigma }_{\infty ,\infty }})\nonumber \\&\lesssim t^{\frac{\sigma +2\beta -1}{2\beta }}\sup _{\frac{t}{4}<\tau <t}\Vert (u_{n}(\tau ),b_{n}(\tau ))\Vert _{B^{-r}_{\infty ,\infty }} \sup _{\frac{t}{4}<\tau <t}\Vert (u_{n}(\tau ),b_{n}(\tau ))\Vert _{B^{\sigma }_{\infty ,\infty }}\nonumber \\&\lesssim t^{-\frac{2\beta -1-r}{2\beta }} h_{n}(-r,\delta )h_{n}(\sigma ,\delta )\nonumber \\&\lesssim t^{-\frac{2\beta -1-r}{2\beta }} h_{n}(-r,\delta )\Theta (\delta )\nonumber \\&\lesssim t^{-\frac{2\beta -1-r}{2\beta }} h_{n}(-r,\delta )\Theta (\delta _{0}), \end{aligned}$$

(4.9)

where we have used the inequality (4.6) in the last inequality. In a similar way, we can estimate $\widetilde{J}_{n}(t)$ as

$$\begin{aligned} \Vert \widetilde{J}_{n}(t)\Vert _{B^{-r}_{\infty ,\infty }}\lesssim&t^{-\frac{2\beta -1-r}{2\beta }} h_{n}(-r,\delta )\Theta (\delta _{0}). \end{aligned}$$

(4.10)

Inserting estimates (4.7)–(4.10) into (4.4) and (4.5), we deduce that there exists a constant $C_{2}>0$ which independent of $t,\delta$ and $n$ such that

$$\begin{aligned} h_{n}(-r,\delta )\le C_{2}\Theta (\delta )+C_{2}\Theta (\delta _{0})h_{n}(-r,\delta ). \end{aligned}$$

Thus, for small enough $\delta _{0}$, we have

$$\begin{aligned} h_{n}(-r,\delta )\le 2C_{2}\Theta (\delta ). \end{aligned}$$

(4.11)

By using the classical interpolation inequality (see [16, 17])

$$\begin{aligned} \Vert f\Vert _{L^{\infty }}\lesssim (\Vert f\Vert _{B^{-r}_{\infty ,\infty }})^{\frac{\sigma }{r+\sigma }}(\Vert f\Vert _{B^{\sigma }_{\infty ,\infty }})^{\frac{r}{r+\sigma }}, \end{aligned}$$

(4.6) and (4.11) imply that there are two constants $C>0$ and $\delta _{0}\in (0.\frac{T}{2}]$ which independent of $n$ such that for all $\delta \in (0,\delta _{0}]$, we have

$$\begin{aligned} \sup _{0<t<\delta }t^{\frac{2\beta -1}{2\beta }}\Vert (u_{n}(t),b_{n}(t))\Vert _{L^{\infty }} \le (h_{n}(-r,\delta ))^{\frac{\sigma }{r+\sigma }}(h_{n}(\sigma ,\delta ))^{\frac{r}{r+\sigma }} \le C\Theta (\delta )<\infty . \end{aligned}$$

This completes the proof of Lemma 4.4.$\square$

We now turn to prove Theorem 4.2.

Proof of Theorem 4.2

We first write $u=u_{1}-u_{2}$ and $b=b_{1}-b_{2}$, then we have

$$\begin{aligned}&u=-B(u,u_{2})-B(u_{2},u)-B(b,b_{1})-B(b_{2},b);\\&b=-B(u,b_{1})-B(u_{2},b)-B(b,u_{1})-B(b_{2},u), \end{aligned}$$

where $B(u,b)$ is defined as $B(u,b):=\int \limits _{0}^{t}e^{-(t-\tau )(-\Delta )^{\beta }} \mathbb P \nabla \cdot (u\otimes b)(\tau )\text{ d}\tau \nonumber$.

Notice that from Propositions 2.4 and 2.7,

$$\begin{aligned} \Vert e^{-(t-\tau )(-\Delta )^{\beta }}\mathbb P \nabla \cdot (u\otimes u_{2})\Vert _{L^{p}}\le C (t-\tau )^{-\frac{1}{2\beta }}\tau ^{-\frac{2\beta -1}{2\beta }}\Vert u\Vert _{L^{p}}\tau ^{\frac{2\beta -1}{2\beta }}\Vert u_{2}\Vert _{L^{\infty }}. \end{aligned}$$

Since

$$\begin{aligned} f\mapsto \int \limits _{0}^{t}(t-\tau )^{-\frac{1}{2\beta }}\tau ^{-\frac{2\beta -1}{2\beta }}f(\tau )\text{ d}\tau \end{aligned}$$

is bounded on $L^{q}$ for $q\in (\frac{2\beta }{2\beta -1},\infty ]$, we get for $t_{0}\in (0,T]$,

$$\begin{aligned} \Vert B(u,u_{2})\Vert _{L^{q}([0,t_{0}];L^{p})}\le C\Vert u\Vert _{L^{q}([0,t_{0}];L^{p})}\sup _{0<\tau <t_{0}}\tau ^{\frac{2\beta -1}{2\beta }}\Vert u_{2}(\tau )\Vert _{L^{\infty }}. \end{aligned}$$

The remainder terms of the right side of (4.10) can be estimated in a similar way. Hence, we have

$$\begin{aligned}&\Vert u\Vert _{L^{q}([0,t_{0}];L^{p})}\nonumber \\&\quad \le \!C \Vert u\Vert _{L^{q}([0,t_{0}];L^{p})} (\!\sup _{0<\tau <t_{0}}\tau ^{\frac{2\beta -1}{2\beta }} \Vert (u_{1}(\tau ),b_{1}(\tau ))\Vert _{L^{\infty }}\nonumber \\&\qquad +\sup _{0<\tau <t_{0}}\tau ^{\frac{2\beta -1}{2\beta }}\Vert (u_{2}(\tau ),b_{2}(\tau ))\Vert _{L^{\infty }}). \end{aligned}$$

The estimate of $b$ can be obtained in a similar way. Hence,

$$\begin{aligned}&\Vert (u,b)\Vert _{L^{q}([0,t_{0}];L^{p})}\nonumber \\&\quad \le C \Vert (u,b)\Vert _{L^{q}([0,t_{0}];L^{p})} (\!\sup _{0<\tau <t_{0}}\tau ^{\frac{2\beta -1}{2\beta }}\Vert (u_{1}(\tau ),b_{1}(\tau ))\Vert _{L^{\infty }}\nonumber \\&\qquad +\sup _{0<\tau <t_{0}}\tau ^{\frac{2\beta -1}{2\beta }}\Vert (u_{2}(\tau ),b_{2}(\tau ))\Vert _{L^{\infty }}). \end{aligned}$$

(4.12)

From (4.12), by using the results of Lemma 4.4, we get that for $t_{0}$ close enough to $0$, $(u,b)\equiv 0$ on $[0,t_{0}]$. Thus, we have local uniqueness on $[0,t_{0}]$. By using the standard continuous argument, this uniqueness can be propagated to the whole $[0,T]$.$\square$