---
abstract: 'We obtain the global well-posedness to the 3D incompressible magnetohydrodynamics (MHD) equations in Besov space with negative index of regularity. Particularly, we can get the global solutions for a new class of large initial data. As a byproduct, this result improves the corresponding result in [@HHW]. In addition, we also get the global result for this system in $\mathcal{\chi}^{-1}({\mathbb{R}}^3)$ originally developed in [@LL]. More precisely, we only assume that the norm of initial data is exactly smaller than the sum of viscosity and diffusivity parameters.'
address: '$^1$ School of Mathematics Sciences, Zhejiang University, Hanzhou 310027, China'
author:
- 'Renhui Wan$^{1}$'
title: 'Global well-posedness to the 3D incompressible MHD equations with a new class of large initial data'
---

.2in

.2in

Introduction {#s1}
============

We are concerned with the 3D incompressible MHD equations: $$\label{1.1}
\left\{
\begin{aligned}
& \partial_t u + u\cdot\nabla u -\mu_1 \Delta u +\nabla p = B\cdot\nabla B,  \\
& \partial_t B + u\cdot\nabla B-B\cdot\nabla u -\mu_2 \Delta B=0,  \\
& {\rm div} u={\rm div}B=0,\\
& u(0,x) =u_0(x),\ B(0,x)=B_0(x),
\end{aligned}
\right.$$ here $(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^3$, $u,p,B$ stand for velocity vector, scalar pressure and magnetic vector, respectively, $\mu_1$ and $\mu_2$ are nonnegative viscosity and diffusivity parameters, respectively.

.1in For $\mu_1>0$ and $\mu_2>0$, the local well-posedness and global existence with small data for (\[1.1\]) were obtained by Duvaut and Lions [@DL] in $d$ dimensional Sobolev space $H^s({\mathbb{R}}^d)$, $s\ge d$. Then Sermange and Termam [@ST] studied the regularity of weak solutions $(u,B)\in L^\infty(0,T;H^1({\mathbb{R}}^3))$. And some regularity criteria were established in [@Wu1; @Wu2; @Wu3]. For $\mu_1>0$ and $\mu_2=0$ (so-called non-resistive MHD equations), by the new Kato-Ponce commutator estimate, $$\|\Lambda^s(u\cdot\nabla B)-u\cdot\nabla \Lambda^s B\|_{L^2({\mathbb{R}}^d)}\le C\|\nabla u\|_{H^s({\mathbb{R}}^d)}\|B\|_{H^s({\mathbb{R}}^d)},\ s>\frac{d}{2},\ d=2,3,$$ Fefferman et al. [@Fefferman] proved the low regularity local well-posedness of strong solutions, which was extended to general inhomogeneous Besov space with initial data $(u_0,B_0)\in B_{2,1}^{\frac{d}{2}-1}({\mathbb{R}}^d)\times B_{2,1}^\frac{d}{2}({\mathbb{R}}^d)$ in the recent works [@Chemin; @1] and [@Wan]. Furthermore, for the non-resistive version with smooth initial data near some nontrivial steady state, we refer [@PZhnag1; @LT; @ZZhang; @PZhang2] for the related works.

.1in Due to a new observation that the velocity field plays a more important role than magnetic field. The new regularity criteria only involving the velocity were proved, see [@CMZ; @HX; @YZ1; @YZ2] and references therein.

.1in One can easily get a new formation of (\[1.1\]) by the following: $$W^+:=u+B,\ W^-:=u-B,\ \nu_{+}=\frac{\mu_1+\mu_2}{2},\ \nu_{-}=\frac{\mu_1-\mu_2}{2}$$ with initial data $W^{\pm}_0(x):=u_0(x)\pm B_0(x),$ that is, $$\label{1.12}
\left\{
\begin{aligned}
& \partial_t W^+ + W^-\cdot\nabla W^+ -\nu_{+}\Delta W^+ +\nabla p = \nu_{-}\Delta W^-,  \\
& \partial_t W^- + W^+\cdot\nabla W^- -\nu_{+}\Delta W^- +\nabla p = \nu_{-}\Delta W^+,  \\
& {\rm div} W^+={\rm div}W^-=0,\\
& W^+(0,x) =W^+_0(x),\ W^-(0,x)=W^-_0(x).
\end{aligned}
\right.$$

Very recently, He et al. [@HHW] obtained the global well-posedness for (\[1.12\]) with initial data $(u_0,B_0)$ satisfying:\
($i$) $\nu_-=0$ and $$\nu_+^{-3}\|W_0^-\|_{L^3}^3\exp\{C\nu_+^{-3}\|W_0^+\|_{L^3}^3\}<\epsilon_0$$ or $$\nu_+^{-3}\|W_0^+\|_{L^3}^3\exp\{C\nu_+^{-3}\|W_0^-\|_{L^3}^3\}<\epsilon_0;$$ ($ii$) $\nu_-\neq0$ and $$\label{1.121}
\left(\nu_+^{-2}\|W^-_0\|_{\dot{H}^\frac{1}{2}}^2+\frac{\nu_-^2}{\nu_+^2}(\nu_+^{-2}\|W_0^+\|_{\dot{H}^\frac{1}{2}}^2+\frac{\nu_-^2}{\nu_+^2})\right)
\exp\left\{C\nu_+^{-4}(\|W_0^+\|_{\dot{H}^\frac{1}{2}}^4+\nu_-^4)\right\}<\epsilon_0$$ or $$\left(\nu_+^{-2}\|W^+_0\|_{\dot{H}^\frac{1}{2}}^2+\frac{\nu_-^2}{\nu_+^2}(\nu_+^{-2}\|W_0^-\|_{\dot{H}^\frac{1}{2}}^2+\frac{\nu_-^2}{\nu_+^2})\right)
\exp\left\{C\nu_+^{-4}(\|W_0^-\|_{\dot{H}^\frac{1}{2}}^4+\nu_-^4)\right\}<\epsilon_0.$$ Here $\epsilon_0$ ia a sufficiently small positive constant. .1in

In this paper, we will prove the global well-posedness of (\[1.1\]) $(\mu_1>0,\mu_2>0)$ in generalized space, $\dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3)$, by make full use of the harmonic analysis tools. The details can be given as follows:

\[t1\] Consider (\[1.1\]) with initial data $(u_0,B_0)\in \dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3),$ $(p,r)\in (1,\infty)\times [1,\infty),$ satisfying ${\rm div}u_0={\rm div}B_0=0$. There exists a constant $C$ and a small constant $\eta>0$ such that if $$\label{1.14}
\left(\|W^-_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}(\nu_-+\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}})^\frac{2} {1-\epsilon}          \right\}<\eta \nu_+$$ or $$\label{1.15}
\left(\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}(\nu_-+\|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}})^\frac{2}{1-\epsilon}           \right\}<\eta\nu_+,$$ where $(\epsilon,r)$ satisfies

$$\label{1.16}
\left\{
\begin{aligned}
 0\le \epsilon<1&, \ {\rm if }\  r=1;  \\
 0< \epsilon<1&,  \ {\rm if }\  1<r\le 2;  \\
1-\frac{2}{r}\le \epsilon<1&,\  {\rm if }\  2<r<\infty.
\end{aligned}
\right.$$

Then (\[1.1\]) admits a unique global solution $(u,B)$ satisfying $$(u,B)\in \tilde{C}([0,\infty); \dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3))\cap \tilde{L}^1([0,\infty); \dot{B}_{p,r}^{\frac{3}{p}+1}({\mathbb{R}}^3)).$$

.1in If $\nu_-=0$, i.e., $\mu_1=\mu_2=\nu_+$, we have a corollary immediately.

\[c1\] Consider (\[1.1\]) with initial data $(u_0,B_0)\in \dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3),$ $(p,r)\in (1,\infty)\times [1,\infty),$ satisfying ${\rm div}u_0={\rm div}B_0=0$. There exists a constant $C$ and a small constant $\eta>0$ such that if $$\label{1.141}
\|W^-_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}^\frac{2}{1-\epsilon} \right\}<\eta \nu_+$$ or $$\label{1.151}
\|W^-_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}^\frac{2}{1-\epsilon} \right\}<\eta \nu_+,$$ where $(\epsilon,r)$ satisfies (\[1.16\]). Then (\[1.1\]) admits a unique global solution $(u,B)$ satisfying $$(u,B)\in \tilde{C}([0,\infty); \dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3))\cap \tilde{L}^1([0,\infty); \dot{B}_{p,r}^{\frac{3}{p}+1}({\mathbb{R}}^3)).$$

\[r1\] ($i$) We will construct the global solution with a new class of large initial data. More precisely, assume that $\phi$ satisfies the condition in Proposition \[l2\], let $$u_0=(\partial_2\phi,-\partial_1\phi,0),\ \ B_0=2\sin^2\frac{x_3}{2\epsilon}(\partial_2\phi,-\partial_1\phi,0),$$ then ${\rm div}u_0={\rm div}B_0=0$ and $\|(u_0,B_0)\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\le \mathfrak{M}$ $(p>3)$, which is independent of $\epsilon$. Moreover, thanks to Proposition \[l2\], there exists a positive constant $C_1$ and $C_2$, $$\|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge C_1,\ \ \|B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge \frac{C_1}{2},$$ $$\|u_0-B_0\|_{\dot{B_{p,r}^{\frac{3}{p}-1}}}\le C_2\epsilon^{1-\frac{3}{p}},$$ which ensures the conditions (\[1.14\])($\nu_+\gg \nu_-$) and (\[1.141\]) hold. Additionally, the assumption $\nu_+\gg\nu_-$ is reasonable in astrophysical magnetic phenomena, see Remark 2.3 in [@HHW]. Combining with the above explanations, this class of large data can lead the global well-posedness to (\[1.1\]).\
($ii$) One can easily check that condition (\[1.14\]) is equal to (\[1.121\]) when $p=r=2$ and choosing $\epsilon=\frac{1}{2}$. By Bernstein’s inequality, we have the following embedding relationship: $$\dot{H}^\frac{1}{2}\hookrightarrow \dot{B}_{p,r}^{\frac{3}{p}-1},\ \ p> 2,  r\ge2.$$ So our result improves the corresponding work under (\[1.121\]) in [@HHW]. By the same way, similar improvements can also be obtained under (\[1.15\]) , (\[1.141\]) and (\[1.151\]).

We shall point out that the above result can not be extended to $p=\infty.$ As a matter of fact, by these works [@ill2] and [@ill1] concerning the well-known Navier-Stokes equations, (\[1.1\]) may ill-posedness in this endpoint Besov space. .1in Next, we consider the space $\chi^{-1}({\mathbb{R}}^3)$, which is smaller than $\dot{B}_{\infty,r}^{-1}$ due to Proposition \[l33\]. It was originally developed in [@LL] and applied to get the global well-posedness for the Navier-Stokes equations under $$\|u_0\|_{\chi^{-1}}<\mu.$$ For MHD equations (\[1.1\]), similar result holds under $$\label{1.2}
\|u_0\|_{\chi^{-1}}+\|B_0\|_{\chi^{-1}}<\min\{\mu,\eta\},$$ see [@KWang] for details. .1in We have some new result in $\chi^{-1}({\mathbb{R}}^3).$

\[t2\] Consider (\[1.1\]) with initial data $(u_0,B_0)\in \chi^{-1}({\mathbb{R}}^3)$ satisfying ${\rm div}u_0={\rm div}B_0=0$. There exists a constant $C$ such that if $$\label{1.21}
\left(\|W_0^-\|_{\chi^{-1}}+\frac{C\nu_-}{\nu_+}(\nu_-+\|W_0^+\|_{\chi^{-1}})\right)\exp\left\{\frac{C}{\nu_+^2}(\nu_-+\|W_0^+\|_{\chi^{-1}})^2\right\}<2\nu_+$$ or $$\label{1.22}
\left(\|W_0^+\|_{\chi^{-1}}+\frac{C\nu_-}{\nu_+}(\nu_-+\|W_0^-\|_{\chi^{-1}})\right)\exp\left\{\frac{C}{\nu_+^2}(\nu_-+\|W_0^-\|_{\chi^{-1}})^2\right\} <2\nu_+.$$ Then (\[1.1\]) admits a unique global solution $(u,B)$ satisfying $$(u,B)\in C([0,\infty); \chi^{-1}({\mathbb{R}}^3))\cap L^1([0,\infty);\chi^1({\mathbb{R}}^3)).$$

Similarly, we also have a corollary immediately when $\nu_-=0$.

\[c2\] Consider (\[1.1\]) with initial data $(u_0,B_0)\in \chi^{-1}({\mathbb{R}}^3)$ satisfying ${\rm div}u_0={\rm div}B_0=0$. There exists a constant $C$ such that if $$\|W_0^-\|_{\chi^{-1}}\exp\left\{\frac{C}{\nu_+^2}\|W_0^+\|_{\chi^{-1}}^2\right\} <2\nu_+$$ or $$\|W_0^+\|_{\chi^{-1}}\exp\left\{\frac{C}{\nu_+^2}\|W_0^-\|_{\chi^{-1}}^2\right\}<2\nu_+.$$ Then (\[1.1\]) admits a unique global solution $(u,B)$ satisfying $$(u,B)\in C([0,\infty); \chi^{-1}({\mathbb{R}}^3))\cap L^1([0,\infty);\chi^1({\mathbb{R}}^3)).$$

\[r2\] The authors in [@LL] proved the global well-posedness for Navier-Stokes equations by using $$\|u\cdot\nabla u\|_{\chi^{-1}}\le \|u\|_{\chi^{-1}}\|u\|_{\chi^1},$$ while we shall use the new estimate below in our proof, i.e., $$\|u\cdot\nabla v\|_{\chi^{-1}}\le \|u\|_{\chi^0}\|v\|_{\chi^0}.$$

\[r3\] Due to the symmetric structure of (\[1.12\]), we only give the proof of Theorem \[t1\] and Theorem \[t2\] under (\[1.14\]) and (\[1.21\]), respectively.

The present paper is structured as follows:\
In section \[s2\], we provide some definitions of spaces, establish several lemmas. The third section proves Theorem \[t1\], while the last section gives the proof of Theorem \[t2\].

.1in Let us complete this section by describing the notations we shall use in this paper.\
[**Notations**]{} The uniform constant $C$ is different on different lines. We also use $L^p$, $\dot{B}_{p,r}^s$ and $\chi^s$ to stand for $L^p(\mathbb{R}^d)$, $\dot{B}_{p,r}^s(\mathbb{R}^d)$ and $\chi^s({\mathbb{R}}^d)$ in somewhere, respectively. We use $A:=B$ to stands for $A$ is defined by $B$, and ${\bf 1}$ is the characteristic function.

.4in

 Preliminaries {#s2}
==============

In this section, we give some necessary definitions, propositions and lemmas. .1in The Fourier transform is given by $$\widehat{f}(\xi)=\int_{{\mathbb{R}}^d}e^{-ix\cdot\xi}f(x)dx.$$

Let $\mathfrak{B}=\{\xi\in\mathbb{R}^d,\ |\xi|\le\frac{4}{3}\}$ and $\mathfrak{C}=\{\xi\in\mathbb{R}^d,\ \frac{3}{4}\le|\xi|\le\frac{8}{3}\}$. Choose two nonnegative smooth radial function $\chi,\ \varphi$ supported, respectively, in $\mathfrak{B}$ and $\mathfrak{C}$ such that $$\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,\ \ \xi\in\mathbb{R}^d\setminus\{0\}.$$ We denote $\varphi_{j}=\varphi(2^{-j}\xi),$ $h=\mathfrak{F}^{-1}\varphi$ and $\tilde{h}=\mathfrak{F}^{-1}\chi,$ where $\mathfrak{F}^{-1}$ stands for the inverse Fourier transform. Then the dyadic blocks $\Delta_{j}$ and $S_{j}$ can be defined as follows $$\Delta_{j}f=\varphi(2^{-j}D)f=2^{jd}\int_{\mathbb{R}^d}h(2^jy)f(x-y)dy,\ \ S_{j}f=\sum_{k\le j-1}\Delta_{k}f$$

Formally, $\Delta_{j}$ is a frequency projection to annulus $\{\xi:\ C_{1}2^j\le|\xi|\le C_{2}2^j\}$, and $S_{j}$ is a frequency projection to the ball $\{\xi:\ |\xi|\le C2^j\}$. One easily verifies that with our choice of $\varphi$ $$\Delta_{j}\Delta_{k}f=0\ {\rm if} \ |j-k|\ge2\ \ {\rm and}\ \  \Delta_{j}(S_{k-1}f\Delta_{k}f)=0\  {\rm if}\  |j-k|\ge5.$$ Let us recall the definition of the Besov space.

\[HB\] Let $s\in \mathbb{R}$, $(p,q)\in[1,\infty]^2,$ the homogeneous Besov space $\dot{B}_{p,q}^s({\mathbb{R}}^d)$ is defined by $$\dot{B}_{p,q}^{s}({\mathbb{R}}^d)=\{f\in \mathfrak{S}'({\mathbb{R}}^d);\ \|f\|_{\dot{B}_{p,q}^{s}({\mathbb{R}}^d)}<\infty\},$$ where $$\|f\|_{\dot{B}_{p,q}^s({\mathbb{R}}^d)}=\left\{\begin{aligned}
&\displaystyle (\sum_{j\in \mathbb{Z}}2^{sqj}\|\Delta_{j}f\|_{L^p({\mathbb{R}}^d)}^{q})^\frac{1}{q},\ \ \ \ {\rm for} \ \ 1\le q<\infty,\\
&\displaystyle \sup_{j\in\mathbb{Z}}2^{sj}\|\Delta_{j}f\|_{L^p({\mathbb{R}}^d)},\ \ \ \ \ \ \ \ {\rm for}\ \ q=\infty,\\
\end{aligned}
\right.$$ and $\mathfrak{S}'({\mathbb{R}}^d)$ denotes the dual space of $\mathfrak{S}({\mathbb{R}}^d)=\{f\in\mathcal{S}(\mathbb{R}^d);\ \partial^{\alpha}\hat{f}(0)=0;\ \forall\ \alpha\in \ \mathbb{N}^d $ [multi-index]{}} and can be identified by the quotient space of $\mathcal{S'}/\mathcal{P}$ with the polynomials space $\mathcal{P}$.

The norm of the space $\tilde{L}^{r_1}_t(\dot{B}_{p,r}^s)$ and $\tilde{L}^{r_1}_{t,\omega}(\dot{B}_{p,r}^s)$ is defined by $$\|f\|_{\tilde{L}^{r_1}_t(\dot{B}_{p,r}^s)}:=\|2^{js}\|\Delta_j f\|_{L^{r_1}_tL^p}\|_{l^r({\mathbb{Z}})}$$ and $$\|f\|_{\tilde{L}^{r_1}_{t,\omega}(\dot{B}_{p,r}^s)}:=\|2^{js}\left(\int_0^t \omega(\tau)^{r_1}\|\Delta_j f(\tau)\|_{L^p}^{r_1} d\tau\right)^\frac{1}{r_1}\|_{l^r({\mathbb{Z}})}.$$ $f\in \tilde{C}(0,t;\dot{B}_{p,r}^s)$ means $f\in \tilde{L}^{\infty}_t(\dot{B}_{p,r}^s)$ and $\|f(t)\|_{\dot{B}_{p,r}^s}$ is continuous in time.

.1in The following proposition provide Bernstein type inequalities.

Let $1\le p\le q\le \infty$. Then for any $\beta,\gamma\in (\mathbb{N}\cup \{0\})^3$, there exists a constant $C$ independent of $f,j$ such that

1.  If $f$ satisfies $$\mbox{supp}\, \widehat{f} \subset \{\xi\in \mathbb{R}^d: \,\, |\xi|
    \le \mathcal{K} 2^j \},$$ then $$\|\partial^\gamma f\|_{L^q(\mathbb{R}^d)} \le C 2^{j|\gamma|  +
    j d(\frac{1}{p}-\frac{1}{q})} \|f\|_{L^p(\mathbb{R}^d)}.$$

2.  If $f$ satisfies $$\label{spp}
    \mbox{supp}\, \widehat{f} \subset \{\xi\in \mathbb{R}^d: \,\, \mathcal{K}_12^j
    \le |\xi| \le \mathcal{K}_2 2^j \}$$ then $$\|f\|_{L^p(\mathbb{R}^d)} \le C2^{-j|\gamma|}\sup_{|\beta|=|\gamma|} \|\partial^\beta f\|_{L^p(\mathbb{R}^d)}.$$

For more details about Besov space such as some useful embedding relations, see [@BCD; @Grafakos; @Stein].

\[l1\][@Danchin] Let $1<p<\infty,$ $supp \widehat{u}\subset C(0,R_{1},R_{2})$ (with $0<R_{1}<R_{2}$). There exists a constant $c$ depending on $\frac{R_{2}}{R_{1}}$ and such that $$\label{Ber}
c\frac{R_{1}^2}{p^2}\int_{\mathbb{R}^3}|u|^p dx\le -\frac{1}{p-1}\int_{\mathbb{R}^3} \Delta u |u|^{p-2}u dx.$$

\[l2\] Let $\phi\in \mathcal{S}({\mathbb{R}}^3)$, whose Fourier transform supported in annulus contained in ${\mathbb{R}}^3\setminus\{0\}$, and $p>3$. If $u_0=(\partial_2 \phi, -\partial_1\phi,0)$ and $B_0=2\sin^2\frac{x_3}{2\epsilon}(\partial_2 \phi, -\partial_1\phi,0),$ then there exists a constant $C_1,C_2>0$ such that $$\|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge C_1,\ \ \|B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge \frac{C_1}{2}$$ and $$\|u_0-B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\le C_2\epsilon^{1-\frac{3}{p}},$$ here $\epsilon$ is sufficiently small.

The last estimate can be obtained by following the proof of Lemma 3.1 in [@Chemin; @2]. So we suffice to show both $\|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}$ and $\|B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}$ has positive lower bound. With this $\phi$, there exists a finite $j_0\in{\mathbb{Z}}$, such that $\Delta_{j_0}\partial_2\phi\neq0$, which implies $$\|\Delta_{j_0} \partial_2\phi\|_{L^\infty}\ge \epsilon_0$$ for some positive constant $\epsilon_0$. Thanks to this, by Bernstein’s inequality, we have $$\|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge \|u_0\|_{\dot{B}_{\infty,\infty}^{-1}}\ge 2^{-j_0}\|\Delta_{j_0}\partial_2\phi\|_{L^\infty}\ge 2^{-j_0}\epsilon_0,$$ and by triangle inequality $$\|B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge \|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}-\|u_0-B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge
2^{-j_0}\epsilon_0-C_2\epsilon^{1-\frac{3}{p}}\ge 2^{-j_0-1}\epsilon_0$$ due to the sufficient small $\epsilon$. Choosing $C_1=2^{-j_{0}}\epsilon_0$ yields the desired result.

For some convenience, we provide the following definition of $\chi^{s}({\mathbb{R}}^d)$, $$\|f\|_{\chi^s}:=\int_{{\mathbb{R}}^d} |\xi|^s |\hat{f}(\xi)| d\xi,$$ and we refer [@LL] for some details.

\[l33\] Let $f\in \chi^{-1}$, then we have $$\|f\|_{\dot{B}_{\infty,r}^{-1}}\le \|f\|_{\dot{B}_{\infty,1}^{-1}}\le \|f\|_{\mathbb{{B}}_{1,1}^{-1}}\thickapprox \|f\|_{\chi^{-1}},$$ where $$\|f\|_{\mathbb{B}_{1,1}^{-1}}:=\sum_{j\in{\mathbb{Z}}}2^{-j}\|\widehat{\Delta_j f}\|_{L^1}.$$

The first inequality is obvious, while the second inequality can be proved by using $\|f\|_{L^\infty}\le \|\hat{f}\|_{L^1}.$ Now, we prove $\|f\|_{\mathbb{{B}}_{1,1}^{-1}}\thickapprox \|f\|_{\chi^{-1}}$. By the definition of $\Delta_j$, and using Monotone Convergence Theorem, $$\begin{aligned}
\|f\|_{\mathbb{{B}}_{1,1}^{-1}}=&\sum_{j\in {\mathbb{Z}}}2^{-j}\| \varphi(2^{-j}\xi)\hat{f}(\xi)\|_{L^1}\\
\thickapprox&
\sum_{j\in {\mathbb{Z}}}\| |\xi|^{-1}\varphi(2^{-j}\xi)\hat{f}(\xi)\|_{L^1}\\
=&\||\xi|^{-1}|\hat{f}(\xi)|\|_{L^1}\\
=&\|f\|_{\chi^{-1}},
\end{aligned}$$ where we have used $\sum_{j\in Z}\varphi(2^{-j}\xi)=1$ and $\varphi\ge 0$.

\[l3\] ($i$) Let $(p,r)\in [1,\infty)\times[1,\infty],$ ${\rm div}u=0$, then $$\label{SKP1}
\|u\cdot\nabla v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\le C\left(\|u\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}+
\|v\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\|u\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\right);$$ ($ii$) Let $(p,r)\in [1,\infty)\times[1,\infty],$ ${\rm div}u=0$, then $$\label{SKP2}
\|u\cdot\nabla v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\le
C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2},$$ where $0<\epsilon<1$ and $f=\|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-1}}^\frac{2}{1-\epsilon}.$ In particular, (\[SKP2\]) also holds when $(\epsilon,r)=(0,1)$ and $f=\|u\|_{\dot{B}_{p,1}^\frac{3}{p}}^2.$

For the proof, we shall use homogeneous Bony’s decomposition: $$uv=T_uv+T_vu+R(u,v),$$ where $$T_uv=\sum_{j\in \mathbb{Z}}S_{j-1}u\Delta_{j}v,\ \ T_vu=\sum_{j\in \mathbb{Z}}\Delta_{j}uS_{j-1}v,\ \
R(u,v)=\sum_{j\in\mathbb{Z}}\Delta_{j}u\tilde{\Delta}_{j}v,$$ here $\tilde{\Delta}_{j}=\Delta_{j-1}+\Delta_{j}+\Delta_{j+1}.$

The estimate of (\[SKP1\]) can be established by using $$\|u\cdot\nabla v\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\le C\{\|u\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\|v\|_{\dot{B}_{p,r}^{\frac{3}{p}+1}}+
\|v\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\|u\|_{\dot{B}_{p,r}^{\frac{3}{p}+1}}\},$$ whose proof is standard. Thus the goal is the estimate of (\[SKP2\]). By homogeneous Bony’s decomposition, $$\label{2.4}
\begin{aligned}
\|u\cdot\nabla v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\le& \|T_{u_i}\partial_i v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}
+\|T_{\partial_i v}u_i\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+\|R(u,\nabla v)\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\\
:=&I_1+I_2+I_3.
\end{aligned}$$ Let $\theta=\frac{1-\epsilon}{2}$, $0<\epsilon<1$. For $I_1$, using Hölder’s inequality and Bernstein’s inequality, $$\begin{aligned}
I_1\le& \left\|2^{j(\frac{3}{p}-1)}\sum_{|k-j|\le 4}\|\Delta_j(S_{k-1}u\cdot\nabla \Delta_k v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}-1)}\|S_{j-1}u\cdot\nabla \Delta_j v\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}-1)}\int_0^t \|S_{j-1}u\|_{L^\infty}\|\nabla \Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}+\epsilon)}\int_0^t \|u\|_{\dot{B}_{\infty,\infty}^{-\epsilon}}\| \Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}+\epsilon)}\int_0^t \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}\| \Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}+1)(1-\theta)}\|\Delta_j v\|_{L^1_tL^p}^{1-\theta}
(\int_0^t 2^{j(\frac{3}{p}-1)} \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}^\frac{1}{\theta}\| \Delta_j v\|_{L^p}d\tau)^\theta\right\|_{l^r({\mathbb{Z}})}\\
\le& C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^{1-\theta}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\theta
= C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2},
\end{aligned}$$ here $f=\|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-1}}^\frac{2}{1-\epsilon}$ and we have used $$\left\|2^{js}\|S_j u\|_{L^p}\right\|_{l^r({\mathbb{Z}})}\approx \|u\|_{\dot{B}_{p,r}^{s}},\ \forall\ s<0.$$ Similarly, for $I_2$, by Hölder’s inequality and Bernstein’s inequality, $$\begin{aligned}
I_2\le& \left\|2^{j(\frac{3}{p}-1)}\sum_{|k-j|\le 4}\|\Delta_j(\Delta_{k}u\cdot\nabla S_{k-1} v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}-1)}\|\Delta_{j}u\cdot\nabla S_{j-1} v\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}-1)}\int_0^t \|\Delta_{j}u\|_{L^p}\|\nabla S_{j-1} v\|_{L^\infty} d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\epsilon-1)}\int_0^t \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}\sum_{j'\le j-2}2^{j'(\frac{3}{p}+1)}\| \Delta_{j'} v\|_{L^p} d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|\sum_{j'\le j-2}2^{(j-j')(\epsilon-1)}\int_0^t \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}2^{j'(\frac{3}{p}+\epsilon)}\| \Delta_{j'} v\|_{L^p} d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C \left\|2^{j(\frac{3}{p}+\epsilon)}\int_0^t \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}\| \Delta_{j} v\|_{L^p} d\tau\right\|_{l^r({\mathbb{Z}})}\\
\end{aligned}$$ where we have used Young’s inequality for series for the last inequality, i.e., $$\left\|\sum_{j'\le j-2}2^{(j-j')(\epsilon-1)}c_{j'}\right\|_{l^r({\mathbb{Z}})}\le C\|2^{j(\epsilon-1)}{\bf 1}_{j\ge 2}\|_{l^1({\mathbb{Z}})}\|c_j\|_{l^r({\mathbb{Z}})}\le C\|c_j\|_{l^r({\mathbb{Z}})}.$$ Following the same argument as $I_1$, one gets $$I_2\le C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2}.$$ Finally, we bound $I_3$. By Bernstein’s inequality, Young’s inequality for series and Hölder’s inequality, we have $$\begin{aligned}
I_3\le& \left\|2^{j(\frac{3}{p}-1)}\sum_{k\ge j-3}\|\Delta_j(\Delta_{k}u\cdot\nabla \tilde{\Delta}_k v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j\frac{3}{p}}\sum_{k\ge j-3}\|\Delta_j(\Delta_{k}u\otimes \tilde{\Delta}_k v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|\sum_{k\ge j-3}2^{(j-k)\frac{3}{p}}2^{k\frac{3}{p}}\|\Delta_j(\Delta_{k}u\otimes \tilde{\Delta}_k v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{k\frac{3}{p}}\int_0^t \|\Delta_k u\|_{L^p}\|\tilde{\Delta}_k v\|_{L^\infty} d\tau \right\|_{l^r({\mathbb{Z}})}\ (p<\infty)\\
\le& C\left\|2^{k(\frac{3}{p}+\epsilon)}\int_0^t \| u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}\|\tilde{\Delta}_k v\|_{L^p} d\tau \right\|_{l^r({\mathbb{Z}})},
\end{aligned}$$ and using the same way as the estimate of $I_1$ derives $$I_3\le C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2}.$$ Plugging the above estimates into (\[2.4\]) leads the desired result (\[SKP2\]).\
In addition, if $r=1$, the estimate of $I_1$ can be replaced as follows: $$\begin{aligned}
I_1\le& C\left\|2^{j(\frac{3}{p}-1)}\int_0^t \|S_{j-1}u\|_{L^\infty}\|\nabla \Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j\frac{3}{p}}\int_0^t \|u\|_{\dot{B}_{p,1}^\frac{3}{p}}\|\Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}+1)\frac{1}{2}}\|\Delta_j v\|_{L^1_tL^p}^\frac{1}{2}(2^{j(\frac{3}{p}-1)}\int_0^t \|u\|_{\dot{B}_{p,1}^\frac{3}{p}}^2\|\Delta_j v\|_{L^p}d\tau)^\frac{1}{2}\right\|_{l^r}\\
\le& C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1}{2},
\end{aligned}$$ here $f=\|v\|_{\dot{B}_{p,1}^\frac{3}{p}}^2$. At the same time, one can get the new estimates of $I_2$ and $I_3$ with the similar procedure. Thus we complete the proof of this lemma.

.3in

Proof of Theorem \[t1\] {#s3}
=======================

As the Remark \[r3\], it suffices to prove the Theorem \[t1\] under (\[1.14\]). One can get the local existence and uniqueness for (\[1.1\]) by using the standard argument on the Navier-Stokes equations, namely, there exists a $T^\star>0$, such that $$(u,B)\in \tilde{C}([0,T^\star); \dot{B}_{p,r}^{\frac{3}{p}-1})\cap \tilde{L}^1([0,T^\star); \dot{B}_{p,r}^{\frac{3}{p}+1}).$$ Since the equivalence between (\[1.1\]) and (\[1.12\]), we will consider (\[1.12\]) and suffice to prove $T^\star=\infty$.\
Now, we begin the proof. Let us consider $0<\epsilon<1$ and $r\le \frac{2}{1-\epsilon}$, containing all cases in (\[1.16\]) except $(\epsilon,r)=(0,1).$ Define $$\label{3.1}
\bar{T}:=\sup\left\{t\in (0,T^\star):\ \|W^-\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+\nu_+\|W^-\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\le \epsilon_o \nu_+ \right\},$$ where $\epsilon_0$ is small positive constant and will be determined later on.\
[**Step 1. The estimate of $W^+$.**]{} Consider the first equation in (\[1.12\]), using (\[Ber\]), we get $$\frac{d}{dt}\|\Delta_j W^+\|_{L^p}+c\nu_+ 2^{2j}\|\Delta_j W^+\|_{L^p}\le C\|\Delta_j (W^-\cdot\nabla W^+)\|_{L^p}+C\nu_- 2^{2j}\|\Delta_j W^-\|_{L^p},$$ which yields by a standard procedure $$\begin{aligned}
\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}&+c\nu_+\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
\le& 2\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\|W^-\cdot\nabla W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}
+C\nu_-\|W^-\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}.
\end{aligned}$$ By (\[SKP1\]) and (\[3.1\]), we have for all $t\in (0,\bar{T}],$ $$\begin{aligned}
\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+&c\nu_+\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
\le 2\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^-\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
&+C(\|W^-\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}
\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}+\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}
\|W^-\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})})\\
\le& 2\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\epsilon_0 ( \nu_+\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}+
\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})})+C\epsilon_0\nu_-,\\
\end{aligned}$$ with the selection of $\epsilon_0<\min\{\frac{c}{2C},\frac{1}{2C}\}$ leads $$\label{3.2}
\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+c\nu_+\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
\le 4\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+2c\nu_-.$$

[**Step 2. The estimate of $W^-$.**]{} Denote $$f(t):=\|W^+(t)\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}^\frac{2}{1-\epsilon},\ W_\lambda^\pm:=W^\pm\exp\{-\lambda\int_0^t f(\tau) d\tau\},
\ p_\lambda:=p\exp\{-\lambda\int_0^t f(\tau) d\tau\},$$ where $\lambda$ is large enough constant and will be determined later on. So we can rewrite the second equation in (\[1.12\]) as $$\partial_t W^-_\lambda+\lambda f(t)W^-_\lambda+W^+\cdot\nabla W^-_\lambda+\nabla p_\lambda-\nu_+\Delta W^-_\lambda=\nu_-\Delta W^+_\lambda.$$ By a similar procedure, we have $$\begin{aligned}
\|\Delta_j W^-_\lambda\|_{L^\infty_t L^p}+&\lambda \int_0^t f(\tau)\|\Delta_j W^-_\lambda\|_{L^p} d\tau+c\nu_+2^{2j}\|\Delta_j W^-_\lambda\|_{L^1_tL^p}\\
&\le \|\Delta_j W_0^-\|_{L^p}+C\|\Delta_j(W^+\cdot\nabla W^-_\lambda)\|_{L^1_tL^p}+C\nu_-2^{2j}\|\Delta_j W^+_\lambda\|_{L^1_tL^p}.
\end{aligned}$$ Then we obtain $$\begin{aligned}
\|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+&c\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+\lambda \|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}\\
&\le \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+C\|W^+\cdot\nabla W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}.
\end{aligned}$$ Thanks to (\[SKP2\]), and by Young’s inequality, we obtain $$\label{3.3}
\begin{aligned}
&\ \ \ \|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+c\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+\lambda \|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}\\
\le& \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+C\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}
\|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2}\\
\le& \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+\frac{c\nu_+}{8}\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
&+C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}}\|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}.
\end{aligned}$$ Choosing $\lambda>2C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}},$ absorbing the third and fourth term on the right hand side of last inequality by the left hand side in (\[3.3\]) follows $$\begin{aligned}
\|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}&+\frac{7c}{8}\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}}\|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}\\
\le& \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}.
\end{aligned}$$ Obviously, using (\[3.2\]), we have $$\begin{aligned}
\|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+c\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
\le&2\|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
\le& C\left( \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right).
\end{aligned}$$ This yields, after using (\[3.2\]) again, for all $t\in(0,\bar{T}),$ $$\begin{aligned}
&\ \ \ \|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+c\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
\le& C\left( \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}}\int_0^t \|W^+(\tau)\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}^\frac{2}{1-\epsilon}d\tau
\right\}\\
\le& C\left( \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}}\|W^+\|_{\tilde{L}^\frac{2}{1-\epsilon}_t(\dot{B}_{p,r}^{\frac{3}{p}-\epsilon})}^\frac{2}{1-\epsilon}\right\}\\
\le& C\left( \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}(\nu_-+\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}})^\frac{2}{1-\epsilon}\right\},
\end{aligned}$$ which implies that if we take $\eta$ small enough in (\[1.14\]), there holds for all $t\le \bar{T}$, $$\|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\le C\eta \nu_+<\frac{\epsilon_0}{2}\nu_+.$$ Then by a standard continuous method, we get $\bar{T}=T^\star=\infty.$ .1in The remainder is $r=1, \epsilon=0$, by a similar arguments, using (\[SKP2\]) for this case and let $(\epsilon,r)=(0,1)$, $f=\|W^+\|_{\dot{B}_{p,1}^\frac{3}{p}}^2$ in (\[3.3\]), the desired result can be otained. Hence, we complete the proof of Theorem \[t1\].

.3in

Proof of Theorem \[t2\] {#s4}
=======================

One can easily get the local well-posedness of (\[1.1\]), that is, there exists a $T^\star>0$ such that $$(u,B)\in C([0,T^\star); \chi^{-1}({\mathbb{R}}^3))\cap L^1([0,T^\star); \chi^1({\mathbb{R}}^3)).$$ So we suffices to show $T^\star=\infty$. .1in Now, we begin the proof. (\[1.21\]) is indeed equal to $$\label{4.1}
\left(\|W_0^-\|_{\chi^{-1}}+\frac{C\nu_-}{\nu_+}(\nu_-+\|W_0^+\|_{\chi^{-1}})\right)\exp\left\{\frac{C}{\nu_+^2}(\nu_-+\|W_0^+\|_{\chi^{-1}})^2\right\}\le(2-\epsilon_0)\nu_+$$ for some $\epsilon_0>0.$ And next we suffices to prove the desired result under (\[4.1\]). Let $C_1,C_2\in (0,2)$ satisfying $2(2-\epsilon_0)^2<C_2(2-C_1)^2$ and $$a=C_2 \nu_+,\ \ a_1=C_1 \nu_+,\ b\in (\frac{2(2-\epsilon_0)}{2-C_1}\nu_+,\sqrt{2C_2}\nu_+).$$ Then consider the first equation in (\[1.12\]), by the procedure as [@LL], and using interpolation inequality, we have $$\begin{aligned}
\frac{d}{dt}\|W^+\|_{\chi^{-1}}+&\nu_+\|W^+\|_{\chi^1}\le \|W^+\cdot\nabla W^-\|_{\chi^{-1}}+\nu_-\|W^-\|_{\chi^1}\\
\le& \|W^+\|_{\chi^0}\|W^-\|_{\chi^0}+\nu_-\|W^-\|_{\chi^1}\\
\le& \|W^-\|_{\chi^0}^2\|W^+\|_{\chi^{-1}}^\frac{1}{2}\|W^+\|_{\chi^1}^\frac{1}{2}+\nu_-\|W^-\|_{\chi^1}\\
\le & \frac{1}{2a}\|W^-\|_{\chi^0}^2\|W^+\|_{\chi^{-1}}+\frac{a}{2}\|W^+\|_{\chi^1}+\nu_- \|W^-\|_{\chi^1},
\end{aligned}$$ which derives by integrating in time, $$\label{4.12}
\begin{aligned}
\|W^+\|_{L^\infty_t(\chi^{-1})}+&(\nu_+-\frac{a}{2})\|W^+\|_{L^1_t(\chi^1)}\\
\le& \frac{1}{2a}\|W^-\|_{L^2_t(\chi^0)}^2\|W^+\|_{L^\infty_t(\chi^{-1})}
+\nu_-\|W^-\|_{L^1_t(\chi^1)}+\|W_0^+\|_{\chi^{-1}}.
\end{aligned}$$ Define $$\label{4.2}
\bar{T}:=\sup\left\{t\in (0,T^\star):\ \|W^-\|_{L^\infty_t(\chi^{-1})}+\nu_+\|W^-\|_{L^1_t(\chi^1)}\le b \right\}.$$ Then we will prove $T^\star=\bar{T}=\infty$ under (\[4.1\]). Using (\[4.2\]), combining with (\[4.12\]), we have $$\label{4.3}
(1-\frac{b^2}{2a\nu_+})\|W^+\|_{L^\infty_t(\chi^{-1})}+(\nu_+-\frac{a}{2})\|W^+\|_{L^1_t(\chi^1)}\le \|W_0^+\|_{\chi^{-1}}+\frac{b\nu_-}{\nu_+}.$$ Following the similar way as (\[4.12\]), one gets $$\frac{d}{dt}\|W^-\|_{\chi^{-1}}+\nu_+\|W^-\|_{\chi^1}
\le \frac{1}{2a_1}\|W^+\|_{\chi^0}^2\|W^-\|_{\chi^{-1}}+\frac{a_1}{2}\|W^-\|_{\chi^1}+\nu_- \|W^+\|_{\chi^1}$$ and thanks to (\[4.3\]), $$\begin{aligned}
\|W^-(t)\|_{\chi^{-1}}+&(\nu_+-\frac{a_1}{2})\|W^-\|_{L^1_t(\chi^1)}\\
\le& \frac{1}{2a_1}\int_0^t \|W^+\|_{\chi^0}^2\|W^-\|_{\chi^{-1}} d\tau
+\frac{\nu_-}{\nu_+-\frac{a}{2}}(\frac{b\nu_-}{\nu_+}+\|W_0^+\|_{\chi^{-1}})+\|W^{-}_0\|_{\chi^{-1}},
\end{aligned}$$ with the application of Gronwall’s lemma, by interpolation’s inequality and (\[4.3\]) leads $$\begin{aligned}
&\ \ \ \ \ \ \|W^-\|_{L^\infty_t(\chi^{-1})}+(\nu_+-\frac{a_1}{2})\|W^-\|_{L^1_t(\chi^1)}\\
\le& (\|W_0^-\|_{\chi^{-1}}+\frac{\nu_-}{\nu_+-\frac{a}{2}}(\frac{b\nu_-}{\nu_+}+\|W_0^+\|_{\chi^{-1}})\exp\left\{\frac{1}{2a_1}\int_0^t \|W^+\|_{\chi^0}^2 d\tau \right\}\\
\le& (\|W_0^-\|_{\chi^{-1}}+\frac{\nu_-}{\nu_+-\frac{a}{2}}(\frac{b\nu_-}{\nu_+}+\|W_0^+\|_{\chi^{-1}})\exp\left\{\frac{1}{2a_1}\|W^+\|_{L^\infty_t(\chi^{-1})}
\|W^+\|_{L^1_t(\chi^1)}\right\}\\
\le& (\|W_0^-\|_{\chi^{-1}}+\frac{\nu_-}{\nu_+-\frac{a}{2}}(\frac{b\nu_-}{\nu_+}+\|W_0^+\|_{\chi^{-1}})\exp\left\{\frac{2a\nu_+}{a_1(2a\nu_+-b^2)(2\nu_+-a_1)}
(\frac{b\nu_-}{\nu_+}+\|W^+_0\|_{\chi^{-1}})^2
 \right\}.
\end{aligned}$$ which indicates that there exists constant $C$ such that $$\begin{aligned}
\|W^-\|_{L^\infty_t(\chi^{-1})}+&(1-\frac{C_1}{2})\nu_+\|W^-\|_{L^1_t(\chi^1)}\\
\le& \left(\|W_0^-\|_{\chi^{-1}}+\frac{C\nu_-}{\nu_+}(\nu_-+\|W_0^+\|_{\chi^{-1}})\right)\exp\left\{\frac{C}{\nu_+^2}(\nu_-+\|W_0^+\|_{\chi^{-1}})^2\right\}\\
\le& (2-\epsilon_0)\nu_+.
\end{aligned}$$ This implies that $$\|W^-\|_{L^\infty_t(\chi^{-1})}+\nu_+\|W^-\|_{L^1_t(\chi^1)}<\frac{2(2-\epsilon_0)}{2-C_1}<b.$$ Therefore, by standard continuous method, we get $T^\star=\bar{T}=\infty.$ This concludes the proof of Theorem \[t2\].

.4in

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