Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations

Abstract

This paper concerns the three-dimensional incompressible generalized magnetohydrodynamic equations. By using the Lei and Lin (Commun Pure Appl Math 64:1297–1304 2011) argument, we get the global well-posedness of the generalized magnetohydrodynamic equations with small initial data. Moreover, we prove that the corresponding global solution decays to zero as time goes to infinity.

1 Introduction

In this paper, we consider the following Cauchy problem of the three-dimensional (3D) generalized incompressible magnetohydrodynamic (GMHD) equations

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}u+(u\cdot \nabla )u+\mu \Lambda ^{2\alpha } u+ \nabla \pi =(B\cdot \nabla ) B,\quad (x,t)\in \mathbb {R}^{3}\times (0,\infty ),\\ \partial _{t}B+(u\cdot \nabla )B +\nu \Lambda ^{2\alpha } B=(B\cdot \nabla )u,\\ \nabla \cdot u=0,\quad \nabla \cdot B=0, \end{array}\right. \end{aligned}$$

(1.1)

with the initial condition

$$\begin{aligned} u(x,0)=u_{0}(x),\,B(x,0)=B_{0}(x),\quad x\in \mathbb {R}^{3}, \end{aligned}$$

where $\mu \ge 0$ and $\nu \ge 0$ are real constant parameters. $u=u(x,t)\in \mathbb {R}^{3}$ denotes the velocity, $\pi =\pi (x,t)\in \mathbb {R}$ denotes scalar pressure, and $B=B(x,t)\in \mathbb {R}^{3}$ is the magnetic field, while $u_{0}(x)$ and $b_{0}(x)$ are the given initial velocity and initial magnetic field with $\nabla \cdot u_{0}(x)=0$ and $\nabla \cdot B_{0}(x)=0$, respectively, while the fractional Laplacian operator $\Lambda ^{2\alpha }$, $\Lambda \triangleq (-\Delta )^{\frac{1}{2}}$ is defined through a Fourier transform, namely $\widehat{\Lambda ^{2\alpha } f}(\xi )=|\xi |^{2\alpha }\hat{f}(\xi )$. The GMHD system (1.1), which describes the macroscopic behavior of the electrically conducting incompressible fluids in a magnetic field, is a generalization of the classical incompressible MHD system.

It is an easy observation that the system (1.1) with $\alpha =1$ reduces to the classical MHD system, which has drawn much attention during the past twenty more years. Duvaut and Lions [5] constructed a global Leray–Hopf weak solution and a local strong solution of the 3D incompressible MHD system. Sermange and Temam [14] further examined the properties of these solutions. But whether this unique local solution can exist globally is an outstanding challenge problem due to the presence of Navier–Stokes equations. For this reason, there are numerous important progresses on the fundamental issue of the blow-up criteria or regularity criteria to the system (1.1) with $\alpha =1$ (see, e.g., [3, 4, 7, 8, 13, 14, 17, 22, 27] and the references cited therein). For the GMHD system (1.1), the global-in-time weak solution was proved by Wu [19]. Moreover, Wu proved that the weak solution becomes the classical solution as long as the power $\alpha \ge \frac{5}{4}$. It is still a big open problem for the power $\alpha <\frac{5}{4}$. For many other interesting results on the GMHD system (1.1), we refer the readers to [10–12, 15, 18, 20, 21, 23–26].

Now let us define the following function space

$$\begin{aligned} \chi ^{s}:=\left\{ f\in \mathcal {D}'\left( \mathbb {R}^{3}\right) \,\,\big |\int _{\mathbb {R}^{3}}{|\xi |^{s}|\widehat{f}(\xi )|\,d\xi } <\infty ,\,\,\,s\in \mathbb {R} \right\} \end{aligned}$$

(1.2)

with the corresponding norm denoted by $\Vert \cdot \Vert _{\chi ^{s}}$.

It should be noted that system (1.1) has scaling property that if (u, B) is a solution of the system (1.1), then for any $\varrho >0$ the functions

$$\begin{aligned} u_{\varrho }(x,t)=\varrho ^{2\alpha -1} u\left( \varrho x, \varrho ^{2\alpha }t\right) , \,\, B_{\varrho }(x,t)=\varrho ^{2\alpha -1} B\left( \varrho x, \varrho ^{2\alpha }t\right) \end{aligned}$$

are also solutions of (1.1) with the corresponding initial data $u_{0,\,\varrho }(x)=\varrho ^{2\alpha -1} u_{0}(\varrho x)$ and $B_{0,\,\varrho }(x)=\varrho ^{2\alpha -1} B_{0}(\varrho x)$. This scaling invariance property is particularly significant and naturally leads to the definition of critical space for the system (1.1). Therefore, it is an obvious fact that

$$\begin{aligned} \Vert u_{\varrho }\Vert _{\chi ^{1-2\alpha }}=\Vert u\Vert _{\chi ^{1-2\alpha }}\quad \text{ and }\quad \Vert B_{\varrho }\Vert _{\chi ^{1-2\alpha }}=\Vert B\Vert _{\chi ^{1-2\alpha }}. \end{aligned}$$

Therefore, $\chi ^{1-2\alpha }$ is the critical space for the system (1.1).

For the small initial data, we will prove the global existence, more precisely

Theorem 1.1

Assume that $(u_{0}, B_{0})\in \chi ^{1-2\alpha }(\mathbb {R}^{3})$ with $\frac{1}{2}\le \alpha \le 1$ obeys the smallness condition

$$\begin{aligned} \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}<\frac{\min \{\mu ,\,\,\nu \}}{2^{2(1-\alpha )}} , \end{aligned}$$

(1.3)

then the system (1.1) admits a unique global solution $(u,\,B)\in C(\mathbb {R}^{+};\,\chi ^{1-2\alpha }(\mathbb {R}^{3}))\cap L^{1}(\mathbb {R}^{+};\,\chi ^{1}(\mathbb {R}^{3}))$. Furthermore, for any $0\le t<\infty$, it holds

$$\begin{aligned}&\Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }}+\left( \mu -2^{2(1-\alpha ) }\left( \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}\right) \right) \int _{0}^{t}{ \Vert u(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\\&\quad +\,\left( \nu -2^{2(1-\alpha ) }(\Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }})\right) \int _{0}^{t}{ \Vert B(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\le \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}. \end{aligned}$$

Remark 1.2

Here we want to emphasized that Lei and Lin [9] proved global well-posedness result with small initial datum in the space $\chi ^{-1}$ for the classical 3D Navier–Stokes equations. Later, Wang and Wang [16] obtained the corresponding result for classical 3D magnetohydrodynamic equations by the argument of Lei and Lin [9]. Therefore, Theorem 1.1 can be viewed as complementary result of works [9] and [16].

Now we state the long-time decay result for the system (1.1).

Theorem 1.3

Let $(u,\,B)\in C(\mathbb {R}^{+};\,\chi ^{1-2\alpha }(\mathbb {R}^{3}))\cap L^{1}(\mathbb {R}^{+};\,\chi ^{1}(\mathbb {R}^{3}))$ with $\frac{5}{6}< \alpha \le 1$ be a global solution of the system (1.1) given by Theorem 1.1, then

$$\begin{aligned} \limsup _{t\rightarrow \infty }(\Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }})=0. \end{aligned}$$

(1.4)

Remark 1.4

At present, we are not able to show that (1.4) still holds true for $\frac{1}{2}\le \alpha \le \frac{5}{6}$. The key reason is that the proof heavily relies on Lemma 3.1.

2 The proof of the Theorem 1.1

This section is devoted to the proof of the Theorem 1.1. Before proving the theorem, we need the following lemma which plays a key role in proving our main result.

Lemma 2.1

Assume that $\frac{1}{2}\le \alpha \le 1$, then the following inequality holds

$$\begin{aligned} |\xi |^{2(1-\alpha )}\le \frac{2^{2(1-\alpha )}}{2} \left( |\eta ||\xi -\eta |^{1-2\alpha }+|\eta |^{1-2\alpha }|\xi -\eta |\right) , \end{aligned}$$

(2.1)

for any $\xi ,\,\eta \in \mathbb {R}^{3}$.

Remark 2.2

It should be noted that inequality (2.1) fails for $\alpha >1$. One can take such a counterexample $\eta =(1,\,0,\,0),\,\,\xi =(1-a,\,0,\,-\sqrt{1-a^{2}})$ for $\frac{1}{2}<a\le 1$. The following counterexample also indicates that inequality (2.1) does not hold true for the case $\alpha <\frac{1}{2}$: $\xi =(1,\,0,\,0),\,\,\eta =(1-b,\,0,\,-\sqrt{1-b^{2}})$ with b close enough to 1.

Proof of Lemma 3.1

The cases $\alpha =\frac{1}{2}$ and $\alpha =1$ are quite obvious. In fact,

$$\begin{aligned} |\xi |\le |\eta |+|\xi -\eta | \end{aligned}$$

gives the result for the case $\alpha =\frac{1}{2}$, while the case $\alpha =1$ is a direct consequence of the following mean value inequality

$$\begin{aligned} \frac{a}{b}+\frac{b}{a}\ge 2,\quad a\ge 0,\,\,b\ge 0. \end{aligned}$$

Thus, in what follows we just consider the remainder case $\frac{1}{2}<\alpha <1$. Without loss of generality, we may assume $\xi \ne 0$ (in fact $\xi =0$ is an ordinary case). In this case, the inequality (2.1) can be rewritten as

$$\begin{aligned} \frac{2}{2^{2(1-\alpha )}}\le \frac{|\eta |}{|\xi |}\left( \frac{|\xi -\eta |}{|\xi |}\right) ^{1-2\alpha } +\left( \frac{|\eta |}{|\xi |}\right) ^{1-2\alpha }\frac{|\xi -\eta |}{|\xi |}. \end{aligned}$$

(2.2)

For notational convenience, we denote

$$\begin{aligned} x=\frac{|\eta |}{|\xi |},\quad y=\frac{|\xi -\eta |}{|\xi |}. \end{aligned}$$

It is easy to find

$$\begin{aligned} x+y\ge 1,\,\,x\ge 0,\,\,y\ge 0. \end{aligned}$$

(2.3)

Equivalently, we only show the following inequality holds true under the condition (2.3)

$$\begin{aligned} \frac{2}{2^{2(1-\alpha )}}\le xy^{1-2\alpha }+x^{1-2\alpha }y. \end{aligned}$$

(2.4)

Now we define

$$\begin{aligned} f(x,y)=xy^{1-2\alpha }+x^{1-2\alpha }y. \end{aligned}$$

Now we take linear function $y=k x$ for $0\le k\le \infty$ to cover all the point pair (x, y) on the ranges (2.3). As a result,

$$\begin{aligned} g(x):=f(x,kx)=\left( k^{1-2\alpha }+k\right) x^{2-2\alpha },\,\,\,x\ge \frac{1}{k+1}. \end{aligned}$$

Next, we will prove that the function g(x) admits the minimal value $g_{\mathrm{small}}$ which is independent of k. More precisely, the value $g_{\mathrm{small}}$ is $\frac{2}{2^{2(1-\alpha )}}$.

Taking the derivative of g(x), we see

$$\begin{aligned} g'(x)=2\left( 1-\alpha \right) \left( k^{1-2\alpha }+k\right) x^{1-2\alpha }\ge 0,\quad 0\le k\le \infty . \end{aligned}$$

Hence, we can easily get that

$$\begin{aligned} g_{\mathrm{mim}}(k)=g(\frac{1}{k+1})=\frac{k^{1-2\alpha }+k}{(k+1)^{2-2\alpha }},\quad 0\le k\le \infty . \end{aligned}$$

Under the case $\frac{1}{2}<\alpha <1$, it is easy to check that

$$\begin{aligned} \lim _{k\rightarrow 0}g_{\mathrm{mim}}(k)=\lim _{k\rightarrow \infty }g_{\mathrm{mim}}(k)=+\infty . \end{aligned}$$

Taking the derivative of $g_{\mathrm{mim}}(k)$, we have

$$\begin{aligned} g_{\mathrm{mim}}'(k)=\frac{1+(2\alpha -1)k-(2\alpha -1+k)k^{-2\alpha }}{(k+1)^{3-2\alpha }}. \end{aligned}$$

Now we take

$$\begin{aligned} h(k):=1+(2\alpha -1)k-(2\alpha -1+k)k^{-2\alpha }. \end{aligned}$$

Again taking the derivative of h(k), it yields

$$\begin{aligned} h'(k)=2\alpha -1+\big ((2\alpha -1)k+2\alpha (2\alpha -1)\big )k^{-2\alpha -1}\ge 0. \end{aligned}$$

Moreover, we can find

$$\begin{aligned} h(1)=0, \end{aligned}$$

which together with the monotonicity of h(k) leads to

$$\begin{aligned} h(k)\le 0,\,\,\,0\le k\le 1;\qquad h(k)\ge 0,\,\,\,1\le k\le \infty , \end{aligned}$$

that is,

$$\begin{aligned} g_{\mathrm{mim}}'(k)\le 0,\,\,\,0\le k\le 1;\qquad g_{\mathrm{mim}}'(k)\ge 0,\,\,\,1\le k\le \infty . \end{aligned}$$

Consequently,

$$\begin{aligned} g_{\mathrm{small}}=g_{\mathrm{mim}}(1)=\frac{2}{2^{2(1-\alpha )}}. \end{aligned}$$

This completes the proof of Lemma 3.1. $\square$

Proof of Theorem 1.1

The existence and uniqueness of local smooth solutions can be done as [9, 16]; thus, we may assume that (u, B) is smooth enough in the interval $[0,\,T)$. We will establish some a priori bounds that will allow us to show $T=\infty$ under the assumption (1.3).

To begin with, we take the Fourier transform of the system (1.1) to get

$$\begin{aligned} \partial _{t}\widehat{u}+\mu |\xi |^{2\alpha } \widehat{u}= & {} - i\xi \widehat{\pi }-i\xi \cdot \int _{\mathbb {R}^{3}}{ \widehat{u}(\eta )\otimes \widehat{u}(\xi -\eta )\,d\eta }\nonumber \\&+\,i\xi \cdot \int _{\mathbb {R}^{3}}{ \widehat{B}(\eta )\otimes \widehat{B}(\xi -\eta )\,d\eta } \end{aligned}$$

(2.5)

and

$$\begin{aligned} \partial _{t}\widehat{B}+\nu |\xi |^{2\alpha } \widehat{B}= & {} -i\xi \cdot \int _{\mathbb {R}^{3}}{ \widehat{u}(\eta )\otimes \widehat{B}(\xi -\eta )\,d\eta }\nonumber \\&+\,i\xi \cdot \int _{\mathbb {R}^{3}}{ \widehat{B}(\eta )\otimes \widehat{u}(\xi -\eta )\,d\eta }. \end{aligned}$$

(2.6)

Taking advantage of (2.5) and (2.6), we deduce that

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}t}\int _{\mathbb {R}^{3}}{(|\xi |^{1-2\alpha } |\widehat{u}|+|\xi |^{1-2\alpha } |\widehat{B}| )(\xi )\,\hbox {d}\xi }+\int _{\mathbb {R}^{3}}{(\mu |\xi |\,|\widehat{u}|+\nu |\xi |\,|\widehat{B}|)\,\hbox {d} \xi }\\&\quad \le \int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}{|\xi |^{2-2\alpha } |\widehat{u}(\eta )| \,|\widehat{u}(\xi -\eta )|\,\hbox {d}\eta \hbox {d}\xi }+\int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}{|\xi |^{2-2\alpha } |\widehat{B}(\eta )| \,|\widehat{B}(\xi -\eta )|\,\hbox {d}\eta \hbox {d}\xi }\\&\qquad +\,\int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}{|\xi |^{2-2\alpha } |\widehat{u}(\eta )|\, |\widehat{B}(\xi -\eta )|\,\hbox {d}\eta \hbox {d}\xi }+\int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}{|\xi |^{2-2\alpha } |\widehat{B}(\eta )|\, |\widehat{u}(\xi -\eta )|\,\hbox {d}\eta \hbox {d}\xi }\\&\quad :=J_{1}+J_{2}+J_{3}+J_{4}. \end{aligned}$$

Applying the inequality (2.1) several times, it turns out that

$$\begin{aligned} J_{1}\le & {} \frac{2^{2(1-\alpha )}}{2} \int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}{ \left( |\eta |\,|\xi -\eta |^{1-2\alpha }+|\eta |^{1-2\alpha }|\xi -\eta |\right) |\widehat{u}(\eta )| \,|\widehat{u}(\xi -\eta )|\,\hbox {d}\eta \hbox {d}\xi }\nonumber \\\le & {} 2^{2(1-\alpha )}\int _{\mathbb {R}^{3}}{ |\xi |^{1-2\alpha }|\widehat{u}(\xi )|\, \hbox {d}\xi }\int _{\mathbb {R}^{3}}{ |\xi |\,|\widehat{u}(\xi )|\, \hbox {d}\xi }, \end{aligned}$$

(2.7)

$$\begin{aligned} J_{2}\le & {} \frac{2^{2(1-\alpha )}}{2} \int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}{ \left( |\eta |\,|\xi -\eta |^{1-2\alpha }+|\eta |^{1-2\alpha }|\xi -\eta |\right) |\widehat{B}(\eta )| \,|\widehat{B}(\xi -\eta )|\,\hbox {d}\eta \hbox {d}\xi }\nonumber \\ {}\le & {} 2^{2(1-\alpha )}\int _{\mathbb {R}^{3}}{ |\xi |^{1-2\alpha }|\widehat{B}(\xi )|\, \hbox {d}\xi }\int _{\mathbb {R}^{3}}{ |\xi |\,|\widehat{B}(\xi )|\, \hbox {d}\xi } \end{aligned}$$

(2.8)

and

$$\begin{aligned} J_{3},\,J_{4}\le & {} \frac{2^{2(1-\alpha )}}{2} \int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}{ \left( |\eta |\,|\xi -\eta |^{1-2\alpha }+|\eta |^{1-2\alpha }|\xi -\eta |\right) |\widehat{u}(\eta )| \,|\widehat{B}(\xi -\eta )|\,\hbox {d}\eta \hbox {d}\xi }\nonumber \\ {}\le & {} \frac{2^{2(1-\alpha )}}{2} \int _{\mathbb {R}^{3}}{ |\xi |^{1-2\alpha }|\widehat{B}(\xi )|\, \hbox {d}\xi }\int _{\mathbb {R}^{3}}{ |\xi |\,|\widehat{u}(\xi )|\, \hbox {d}\xi }\nonumber \\&+\,\frac{2^{2(1-\alpha )}}{2} \int _{\mathbb {R}^{3}}{ |\xi |^{1-2\alpha }|\widehat{u}(\xi )|\, \hbox {d}\xi }\int _{\mathbb {R}^{3}}{ |\xi |\,|\widehat{B}(\xi )|\, \hbox {d}\xi }. \end{aligned}$$

(2.9)

Combining aforementioned estimates, we arrive at

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}t}(\Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }}) +\mu \Vert u\Vert _{\chi ^{1}} +\nu \Vert B\Vert _{\chi ^{1}}\nonumber \\&\quad \le 2^{2(1-\alpha )}(\Vert u\Vert _{\chi ^{1-2\alpha }}+\Vert B\Vert _{\chi ^{1-2\alpha }}) (\Vert u\Vert _{\chi ^{1}} +\Vert B\Vert _{\chi ^{1}}). \end{aligned}$$

(2.10)

Note that the condition (1.3), namely

$$\begin{aligned} \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}<\frac{\min \{\mu ,\,\,\nu \}}{2^{2(1-\alpha )}}, \end{aligned}$$

one may deduce that

$$\begin{aligned} \Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }}\le \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}<\frac{\min \{\mu ,\,\,\nu \}}{2^{2(1-\alpha )}}, \end{aligned}$$

(2.11)

for all $t\in [0,\,T)$.

Indeed, we argue (2.11) by contradiction. Suppose that (2.11) is not true. Then there exists a time $T_{\star }<T$ (the first time) such that

$$\begin{aligned} \Vert u(T_{\star })\Vert _{\chi ^{1-2\alpha }}+\Vert B(T_{\star })\Vert _{\chi ^{1-2\alpha }}=\frac{\min \{\mu ,\,\,\nu \}}{2^{2(1-\alpha )}}. \end{aligned}$$

(2.12)

Thus, we have

$$\begin{aligned} \Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }} \le \frac{\min \{\mu ,\,\,\nu \}}{2^{2(1-\alpha )}},\qquad 0\le t\le T_{\star }, \end{aligned}$$

(2.13)

which together with (2.10) leads to

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}t}(\Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }}) +\mu \Vert u\Vert _{\chi ^{1}} +\nu \Vert B\Vert _{\chi ^{1}}\nonumber \\&\quad \le \min \{\mu ,\,\,\nu \} (\Vert u\Vert _{\chi ^{1}} +\Vert B\Vert _{\chi ^{1}}),\qquad 0\le t\le T_{\star }. \end{aligned}$$

(2.14)

Whence we obtain

$$\begin{aligned} \Vert u(T_{\star })\Vert _{\chi ^{1-2\alpha }}+\Vert B(T_{\star })\Vert _{\chi ^{1-2\alpha }}\le \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}<\frac{\min \{\mu ,\,\,\nu \}}{2^{2(1-\alpha )}}, \end{aligned}$$

which leads a contradiction with (2.12). In other words, (2.11) is true.

It follows from (2.10) that for all $t\in [0,\,T)$

$$\begin{aligned}&\Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }} +\left( \mu -2^{2(1-\alpha ) }(\Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }})\right) \int _{0}^{t}{ \Vert u(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\nonumber \\&\quad +\,\left( \nu -2^{2(1-\alpha ) }(\Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }})\right) \int _{0}^{t}{ \Vert B(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\le \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }},\nonumber \\ \end{aligned}$$

(2.15)

It is straightforward to show that

$$\begin{aligned} \Vert \nabla u\Vert _{L^{\infty }}\le \int _{\mathbb {R}^{3}}{ |\xi ||\widehat{u}(\xi )|\, \hbox {d}\xi }\le \Vert u\Vert _{\chi ^{1}} \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla B\Vert _{L^{\infty }}\le \int _{\mathbb {R}^{3}}{ |\xi ||\widehat{B}(\xi )|\, \hbox {d}\xi }\le \Vert B\Vert _{\chi ^{1}}. \end{aligned}$$

Thus, we can deduce from (2.15) that

$$\begin{aligned}&\int _{0}^{T}{\Vert \nabla u(t)\Vert _{L^{\infty }}\,\hbox {d}t}\le \frac{\Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}}{\mu -2^{2(1-\alpha ) }(\Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }})},\\&\int _{0}^{T}{\Vert \nabla B(t)\Vert _{L^{\infty }}\,\hbox {d}t}\le \frac{\Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}}{\nu -2^{2(1-\alpha ) }(\Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }})}. \end{aligned}$$

By the standard energy method, we can show that for any $s>0$

$$\begin{aligned} \Vert u(t)\Vert _{H^{s}}+\Vert B(t)\Vert _{H^{s}}\le & {} (\Vert u_{0}\Vert _{H^{s}}+\Vert B_{0}\Vert _{H^{s}})\text{ exp }\left[ C\int _{0}^{T}{(\Vert \nabla u\Vert _{L^{\infty }}+\Vert \nabla B\Vert _{L^{\infty }})(t)\,\hbox {d}t}\right] \\\le & {} C\text{ exp }\left[ \frac{C(\Vert u_{0}\Vert _{\chi ^{1-2\alpha }} +\Vert B_{0}\Vert _{\chi ^{1-2\alpha }})}{\min \{\mu ,\,\,\nu \}-2^{2(1-\alpha ) }(\Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }})}\right] \\< & {} \infty . \end{aligned}$$

The above estimate implies that $T=\infty$. As a consequence, the following bound holds for any $0\le t<\infty$

$$\begin{aligned}&\Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }} +\left( \mu -2^{2(1-\alpha ) }\left( \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}\right) \right) \int _{0}^{t}{ \Vert u(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\\&\quad +\,\left( \nu -2^{2(1-\alpha ) }\left( \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}\right) \right) \int _{0}^{t}{ \Vert B(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\le \Vert u_{0}\Vert _{\chi ^{1-2\alpha }}+\Vert B_{0}\Vert _{\chi ^{1-2\alpha }}. \end{aligned}$$

Consequently, this completes the proof of Theorem 1.1. $\square$

3 The proof of the Theorem 1.3

Proof of Theorem 1.3

The proof is largely based on the idea from the work [6] (see also [1, 2]).

We take sufficiently small $\epsilon >0$ such that

$$\begin{aligned} \epsilon \le \frac{\min \{\mu ,\,\,\nu \}}{2}. \end{aligned}$$

Now we choose $\widetilde{u_{0}}\in L^{2}$ and $\widetilde{B_{0}}\in L^{2}$ such that

$$\begin{aligned} \Vert \widetilde{u_{0}}\Vert _{\chi ^{1-2\alpha }} +\Vert \widetilde{B_{0}}\Vert _{\chi ^{1-2\alpha }}<\frac{\epsilon }{2}. \end{aligned}$$

(3.1)

This condition is easy to satisfy. As a matter of fact, we select

$$\begin{aligned} \widetilde{u_{0}}=\mathcal {F}^{-1}\left( \widehat{u_{0}}(\xi )I_{|\xi |\le \delta }\right) \quad \text{ and }\quad \widetilde{B_{0}}=\mathcal {F}^{-1}\left( \widehat{B_{0}}(\xi )I_{|\xi |\le \delta }\right) \end{aligned}$$

for small enough $\delta >0$.

Now we consider

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}\widetilde{u}+\left( \widetilde{u}\cdot \nabla \right) \widetilde{u}+\mu \Lambda ^{2\alpha } \widetilde{u}+ \nabla \widetilde{\pi }=\left( \widetilde{B}\cdot \nabla \right) \widetilde{B},\\ \partial _{t}\widetilde{B}+\left( \widetilde{u}\cdot \nabla \right) \widetilde{B} +\nu \Lambda ^{2\alpha } \widetilde{B}=\left( \widetilde{B}\cdot \nabla \right) \widetilde{u},\\ \nabla \cdot \widetilde{u}=0,\quad \nabla \cdot \widetilde{B}=0,\\ \widetilde{u}(x,0)=\widetilde{u_{0}}(x),\quad \widetilde{B}(x,0)=\widetilde{B_{0}}(x). \end{array} \right. \end{aligned}$$

(3.2)

Note that the small condition (3.1); one can deduce from Theorem 1.1 that the system (3.2) has a unique global solution $(\widetilde{u},\,\widetilde{B})\in C\left( \mathbb {R}^{+};\,\chi ^{1-2\alpha }(\mathbb {R}^{3})\right) \cap L^{1}\left( \mathbb {R}^{+};\,\chi ^{1}(\mathbb {R}^{3})\right)$. Moreover, the following holds for any $t\ge 0$

$$\begin{aligned}&\Vert \widetilde{u}(t)\Vert _{\chi ^{1-2\alpha }}+\Vert \widetilde{B}(t)\Vert _{\chi ^{1-2\alpha }}+\frac{\mu }{2}\int _{0}^{t}{ \Vert \widetilde{u}(s)\Vert _{\chi ^{1}}\,\hbox {d}s}+\frac{\nu }{2}\int _{0}^{t}{ \Vert \widetilde{B}(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\\&\quad \le \Vert \widetilde{u_{0}} \Vert _{\chi ^{1-2\alpha }}+\Vert \widetilde{B_{0}}\Vert _{\chi ^{1-2\alpha }}. \end{aligned}$$

Meanwhile, we have the energy estimate

$$\begin{aligned} \Vert \widetilde{u}(t)\Vert _{L^{2}}^{2}+\Vert \widetilde{B}(t)\Vert _{L^{2}}^{2}+2\int _{0}^{t}{(\Vert \Lambda ^{\alpha } \widetilde{u} \Vert _{L^{2}}^{2}+\Vert \Lambda ^{\alpha } \widetilde{B}\Vert _{L^{2}}^{2})(\tau )\,\hbox {d}\tau }\le \Vert \widetilde{u_{0}}\Vert _{L^{2}}^{2}+\Vert \widetilde{B_{0}}\Vert _{L^{2}}^{2}. \end{aligned}$$

Next, we take into consideration the difference

$$\begin{aligned} W=u-\widetilde{u},\quad E=B-\widetilde{B},\quad P=\pi -\widetilde{\pi }, \end{aligned}$$

which satisfies the system

$$\begin{aligned}&\partial _{t}W+(u\cdot \nabla )W+ (W\cdot \nabla )\widetilde{u}+\mu \Lambda ^{2\alpha } W +\nabla P=(B\cdot \nabla ) E+(E\cdot \nabla ) \widetilde{B}, \end{aligned}$$

(3.3)

$$\begin{aligned}&\partial _{t}E+(u\cdot \nabla )E+(W\cdot \nabla )\widetilde{B}+\nu \Lambda ^{2\alpha } E=(B\cdot \nabla )W+(E\cdot \nabla )\widetilde{u}. \end{aligned}$$

(3.4)

Adding the inner products of (3.3) with W and of (3.4) with B and integrating by parts, we can show that

$$\begin{aligned}&\frac{1}{2}\frac{\hbox {d}}{\hbox {d}t}(\Vert W(t)\Vert _{L^{2}}^{2}+\Vert E(t)\Vert _{L^{2}}^{2}) +\mu \Vert \Lambda ^{\alpha } W\Vert _{L^{2}}^{2}+\nu \Vert \Lambda ^{\alpha } E \Vert _{L^{2}}^{2}\nonumber \\\le & {} \left| \int _{\mathbb {R}^{3}}{ (W\cdot \nabla )\widetilde{u}\cdot W\, \hbox {d} x}\right| +\left| \int _{\mathbb {R}^{3}}{ (E\cdot \nabla ) \widetilde{B}\cdot W\, \hbox {d} x}\right| +\left| \int _{\mathbb {R}^{3}}{ (W\cdot \nabla )\widetilde{B}\cdot E\, \hbox {d} x}\right| \nonumber \\&\quad +\,\left| \int _{\mathbb {R}^{3}}{ (E\cdot \nabla )\widetilde{u}\cdot E\, \hbox {d} x}\right| \nonumber \\:= & {} K_{1}+K_{2}+K_{3}+K_{4}, \end{aligned}$$

(3.5)

where we have used the cancellation property

$$\begin{aligned} \int _{\mathbb {R}^{3}}{ (B\cdot \nabla )E\cdot W\, \hbox {d} x}+\int _{\mathbb {R}^{3}}{ (B\cdot \nabla )W\cdot E\, \hbox {d} x}=0. \end{aligned}$$

It follows from the incompressible condition and Young inequality that

$$\begin{aligned} K_{1}\le & {} \left| \int _{\mathbb {R}^{3}}{W \,\widetilde{u} \,\nabla W\, \hbox {d} x}\right| \nonumber \\\le & {} \Vert W\Vert _{L^{2}}\Vert \widetilde{u} \,\nabla W\Vert _{L^{2}}\nonumber \\\le & {} \Vert W\Vert _{L^{2}}\Vert \mathcal {F}(\widetilde{u} \,\nabla W)\Vert _{L^{2}}\nonumber \\\le & {} \Vert W\Vert _{L^{2}}\Vert \widehat{\widetilde{u}}*\widehat{\nabla W}\Vert _{L^{2}}\nonumber \\\le & {} \Vert W\Vert _{L^{2}}\Vert \widehat{\widetilde{u}}\Vert _{L^{2}}\Vert \widehat{\nabla W}\Vert _{L^{1}}\nonumber \\\le & {} \Vert W\Vert _{L^{2}}\Vert \widetilde{u}\Vert _{L^{2}}\Vert W\Vert _{\chi ^{1}}. \end{aligned}$$

(3.6)

The other terms can be bounded as follows by an argument similar to that used in the proof of $K_{1}$

$$\begin{aligned}&K_{2} \le \left| \int _{\mathbb {R}^{3}}{ E\,\widetilde{B} \,\nabla W\, \hbox {d} x}\right| \le \Vert E\Vert _{L^{2}}\Vert \widetilde{B}\Vert _{L^{2}}\Vert W\Vert _{\chi ^{1}}, \end{aligned}$$

(3.7)

$$\begin{aligned}&K_{3} \le \left| \int _{\mathbb {R}^{3}}{W\, \widetilde{B}\,\nabla E\, d x}\right| \le \Vert W\Vert _{L^{2}}\Vert \widetilde{B}\Vert _{L^{2}}\Vert E\Vert _{\chi ^{1}} \end{aligned}$$

(3.8)

and

$$\begin{aligned} K_{4} \le \left| \int _{\mathbb {R}^{3}}{E\,\widetilde{u} \,\nabla E\, \hbox {d} x}\right| \le \Vert E\Vert _{L^{2}}\Vert \widetilde{u}\Vert _{L^{2}}\Vert E\Vert _{\chi ^{1}}. \end{aligned}$$

(3.9)

Plugging the estimates (3.6)–(3.9) into (3.5), it gives

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}t}(\Vert W(t)\Vert _{L^{2}}^{2}+\Vert E(t)\Vert _{L^{2}}^{2}) +\mu \Vert \Lambda ^{\alpha } W\Vert _{L^{2}}^{2}+\nu \Vert \Lambda ^{\alpha } E \Vert _{L^{2}}^{2}\\&\quad \le (\Vert \widetilde{u}\Vert _{L^{2}}\Vert W\Vert _{\chi ^{1}} +\Vert \widetilde{B}\Vert _{L^{2}}\Vert E\Vert _{\chi ^{1}})\Vert W\Vert _{L^{2}}+ (\Vert \widetilde{B}\Vert _{L^{2}}\Vert W\Vert _{\chi ^{1}} +\Vert \widetilde{u}\Vert _{L^{2}}\Vert E\Vert _{\chi ^{1}}) \Vert E\Vert _{L^{2}}. \end{aligned}$$

It should be noted that

$$\begin{aligned} \int _{0}^{t}{ \Vert W(s)\Vert _{\chi ^{1}}\,\hbox {d}s}+\int _{0}^{t}{ \Vert E(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\le C<\infty . \end{aligned}$$

Gronwall inequality allows us to get

$$\begin{aligned} \Vert W\Vert _{L^{2}}^{2}+\Vert E\Vert _{L^{2}}^{2}+\int _{0}^{t}{(\mu \Vert \Lambda ^{\alpha } W\Vert _{L^{2}}^{2}+\nu \Vert \Lambda ^{\alpha } E \Vert _{L^{2}}^{2})(s)\,\hbox {d}s}\le C<\infty . \end{aligned}$$

Now we recall the following interpolation inequality (the proof of which is postponed in the end of this section)

Lemma 3.1

Suppose that $f,\,\,\Lambda ^{\alpha } f\in L^{2}(\mathbb {R}^{n})$ with $\frac{n+2}{6}<\alpha < \frac{n+2}{4}$, then it holds

$$\begin{aligned} \Vert f\Vert _{\chi ^{1-2\alpha }(\mathbb {R}^{3})}\le C \Vert f\Vert _{L^{2}(\mathbb {R}^{n})}^{\frac{6\alpha -2-n}{2\alpha }} \Vert \Lambda ^{\alpha } f\Vert _{L^{2}(\mathbb {R}^{n})}^{\frac{2+n-4\alpha }{2\alpha }}, \end{aligned}$$

where n is the space dimension.

We continue to prove Theorem 1.3. Thanks to above interpolation inequality, one can conclude

$$\begin{aligned}&\int _{0}^{\infty }{\left( \Vert W\Vert _{\chi ^{1-2\alpha }}^{\frac{4\alpha }{5-4\alpha }} +\Vert E\Vert _{\chi ^{1-2\alpha }}^{\frac{4\alpha }{5-4\alpha }}\right) (t)\,\hbox {d}t}\\&\quad \le \int _{0}^{\infty }{\left( \Vert W\Vert _{L^{2}}^{\frac{2(6\alpha -5)}{5-4\alpha }} \Vert \Lambda ^{\alpha } W\Vert _{L^{2}}^{2}+\Vert E\Vert _{L^{2}}^{\frac{2(6\alpha -5)}{5-4\alpha }} \Vert \Lambda ^{\alpha } E\Vert _{L^{2}}^{2}\right) (t)\,\hbox {d}t}\le C<\infty . \end{aligned}$$

Due to the continuity of $W,\,E$ in space $\chi ^{1-2\alpha }$, there exists a time $T_{\star }$ such that

$$\begin{aligned} \Vert W(T_{\star })\Vert _{\chi ^{1-2\alpha }}+\Vert E(T_{\star })\Vert _{\chi ^{1-2\alpha }}<\frac{\epsilon }{2}. \end{aligned}$$

Recalling (3.1), we have

$$\begin{aligned}&\Vert u(T_{\star })\Vert _{\chi ^{1-2\alpha }}+\Vert B(T_{\star })\Vert _{\chi ^{1-2\alpha }}\nonumber \\&\quad \le \, \Vert W(T_{\star })\Vert _{\chi ^{1-2\alpha }}+\Vert E(T_{\star })\Vert _{\chi ^{1-2\alpha }}+ \Vert \widetilde{u}(T_{\star })\Vert _{\chi ^{1-2\alpha }}+\Vert \widetilde{B} (T_{\star })\Vert _{\chi ^{1-2\alpha }}\nonumber \\&\quad < \frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon . \end{aligned}$$

(3.10)

Now we consider the following system

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}u+(u\cdot \nabla )u+\mu \Lambda ^{2\alpha } u+ \nabla \pi =(B\cdot \nabla ) B,\\ \partial _{t}B+(u\cdot \nabla )B +\nu \Lambda ^{2\alpha } B=(B\cdot \nabla )u,\\ \nabla \cdot u=0,\quad \nabla \cdot B=0,\\ {u}(x,0)=u(T_{\star }),\quad {B}(x,0)=W(T_{\star }). \end{array} \right. \end{aligned}$$

By the small data result of Theorem 1.1, we immediately obtain

$$\begin{aligned} \Vert u(t)\Vert _{\chi ^{1-2\alpha }}+\Vert B(t)\Vert _{\chi ^{1-2\alpha }}+(\mu -\epsilon ) \int _{0}^{t}{\Vert u(s)\Vert _{\chi ^{1}}\,\hbox {d}s}+(\nu -\epsilon )\int _{0}^{t}{ \Vert B(s)\Vert _{\chi ^{1}}\,\hbox {d}s}\le \epsilon , \end{aligned}$$

for any $t\ge T_{\star }$.

Due to the arbitrariness of $\epsilon$, the proof of Theorem 1.3 is completed.

Finally, let us finish the proof of Lemma 3.1 to end this section.

Let M be a positive constant to be fixed later.

$$\begin{aligned} \Vert f\Vert _{\chi ^{1-2\alpha }(\mathbb {R}^{n})}\le \int _{|\xi |\le M}{ |\xi |^{1-2\alpha }|\widehat{f}(\xi )| \hbox {d} \xi }+\int _{|\xi |\ge M}{ |\xi |^{1-2\alpha }|\widehat{f}(\xi )| \hbox {d} \xi }:=N_{1}+N_{2}. \end{aligned}$$

Making use of Young inequality, we get

$$\begin{aligned} N_{1}\le & {} \left( \int _{|\xi |\le M}{ |\xi |^{2(1-2\alpha )} \hbox {d} \xi }\right) ^{\frac{1}{2}} \left( \int _{|\xi |\le M}{ |\widehat{f}(\xi )|^{2} \hbox {d} \xi }\right) ^{\frac{1}{2}}\\\le & {} C\left( \int _{0}^{M}{ r^{2(1-2\alpha )}r^{n-1} \hbox {d} r}\right) ^{\frac{1}{2}} \Vert f\Vert _{L^{2}(\mathbb {R}^{n})}\\\le & {} C(\alpha )M^{\frac{2+n-4\alpha }{2}}\Vert f\Vert _{L^{2}(\mathbb {R}^{n})},\quad \left( \alpha < \frac{n+2}{4}\right) . \end{aligned}$$

With the same argument to $N_{2}$, one arrives at

$$\begin{aligned} N_{2}= & {} \int _{|\xi |\ge M}{ |\xi |^{1-3\alpha }(|\xi |^{\alpha }|\widehat{f}(\xi )|) \hbox {d} \xi }\\\le & {} \left( \int _{|\xi |\ge M}{ |\xi |^{2(1-3\alpha )} \hbox {d} \xi }\right) ^{\frac{1}{2}} \left( \int _{|\xi |\ge M}{ |\xi |^{2\alpha }|\widehat{f}(\xi )|^{2} \hbox {d} \xi }\right) ^{\frac{1}{2}}\\\le & {} \left( \int _{M}^{\infty }{ r^{2(1-3\alpha )}r^{n-1} \hbox {d} r}\right) ^{\frac{1}{2}} \Vert \Lambda ^{\alpha }f\Vert _{L^{2}(\mathbb {R}^{n})}\\\le & {} C(\alpha )M^{\frac{2+n-6\alpha }{2}}\Vert \Lambda ^{\alpha }f\Vert _{L^{2}(\mathbb {R}^{n})},\quad \left( \alpha > \frac{n+2}{6}\right) . \end{aligned}$$

Now we select

$$\begin{aligned} M=\left( \frac{\Vert \Lambda ^{\alpha }f\Vert _{L^{2}(\mathbb {R}^{n})}}{\Vert f\Vert _{L^{2}(\mathbb {R}^{n})}}\right) ^{\frac{1}{\alpha }}. \end{aligned}$$

Thus, Lemma 3.1 is now proved. $\square$