Regularity criteria for weak solutions of the Magneto-micropolar equations

1.   Introduction

In this paper we present some regularity criteria for weak solutions of the following magneto-micropolar fluid system in three space dimensions:

where $\mathbf{u}(x,t) = (u_{1}(x,t),u_{2}(x,t),u_{3}(x,t))\in \mathbb{R}^3$ denotes the incompressible velocity field, $\mathbf{w}(x,t) = (w_{1}(x,t),w_{2}(x,t),$ $w_{3}(x,t))\in \mathbb{R}^3$ denotes the micro-rotational velocity, $\mathbf{b}(x,t)$ $= (b_{1}(x,t),b_{2}(x,t),$ $b_{3}(x,t))$ $\in \mathbb{R}^3$ the divergence free magnetic field and $p(x,t)\in \mathbb{R}$ the hydrostatic pressure. As usual, $x\in \mathbb{R}^3$ denotes the space variable and $0 \leq t <T$ the time variable. The positive constants $\mu, \chi, \nu,\kappa$, and $\gamma$ are associated with specific properties of the fluid; precisely, $\mu$ is the kinematic viscosity, $\chi$ is the vortex viscosity, $\kappa$ and $\gamma$ are spin viscosities and, lastly, $\nu^{-1}$ is the magnetic Reynolds number. The initial data for the velocity and magnetic fields, given by $\mathbf{u}_{0}$ and $\mathbf{b}_{0}$ in (1), are divergence free, i.e., $\nabla\cdot\:\mathbf{u}_{0} = \nabla\cdot\:\mathbf{b}_{0} = 0.$

Let us list some recent papers which discuss regularity of weak solutions of the magneto-micropolar equations (1) and systems that are particular cases of these equations as, for example, the classical Navier-Stokes equations.

In $2010$, Y. Baoquan [18] established the following regularity criteria: a solution $(\mathbf{u},\mathbf{w},\mathbf{b})(t)$ of (1) can be extended smoothly beyond $t = T$ if

or

provided that the initial data $(\mathbf{u}_0,\mathbf{w}_0,\mathbf{b}_0)$ belong to $H^1(\mathbb{R}^3)$ and $\nabla \cdot \mathbf{u}_0 = \nabla \cdot \mathbf{b}_0 = 0.$ Here $(\mathbf{u},\mathbf{w},\mathbf{b})\in C([0,T);H^1(\mathbb{R}^3))\cap C((0,T);H^2(\mathbb{R}^3))$ is assumed. Notice that Y. Baoquan [18] used only the velocity field or its gradient in the regularity criteria described above.

In 2013, Y. Wang [14] showed that a weak solution $(\mathbf{u},\mathbf{w},\mathbf{b})(t)$ of (1), defined on the interval $[0,T)$, can be extended smoothly beyond $t = T$ if one assumes

provided that $(\mathbf{u}_0,\mathbf{w}_0,\mathbf{b}_0)\in H^1(\mathbb{R}^3)$ and $\nabla \cdot \mathbf{u}_0 = \nabla \cdot \mathbf{b}_0 = 0.$ Let us point out that Y. Wang [14] established the regularity criterion given above by considering only one component of the gradient of the velocity field (see (2)). The paper [14] deals with an extension of a regularity criterion obtained for weak solutions of the magneto-hydrodynamic equations (see [2]). This latter system is the special case of(1) obtained for $\mathbf{w} = 0$ and $\chi = 0$. Further results related to the current paper are in [2,21].

The papers [14] and [18] (see also [3,8,12,13,15,17,19,20,22,23,24,28,29]), raised our interest to obtain regularity criteria for weak solutions of the magneto-micropolar system (1), which involve only one component of the velocity field $\mathbf{u}$.

Note that the magneto-micropolar system $(1)$ reduces to the classical Navier-Stokes equations for the velocity field $\mathbf{u}$ and the pressure $p$ if one assumes that $\mathbf{w} = \mathbf{b} = 0$ and $\chi = 0.$ Regularity criteria for weak solutions of Navier-Stokes equations, involving only one component of the velocity field, have been published recently. We want to comment on two results.

First, Z. Zhang and X. Yang [22] present a regularity criterion for the Navier-Stokes equations involving the gradient of one component of the velocity field. Precisely, if $\mathbf{u}(t)$ is a weak solution of the Navier-Stokes equations in $(0,T)$ and

then $\mathbf{u}$ is $C^\infty$ in $\mathbb{R}^3\times (0,T)$ and the solution can be extended as a $C^\infty$ function beyond $T$. Here $\mathbf{u}_0\in L^2(\mathbb{R}^3)$ with $\nabla \cdot \mathbf{u}_0 = 0$ is assumed. In the current paper we prove that the criterion (3) can be extended from the Navier-Stokes equations to the magneto-micropolar system (1) under appropriate conditions on the other fields $\mathbf{w}$ and $\mathbf{b}$. See Theorem 1.1 below.

The second paper we cite is [23]. Z. Zhang and X. Yang [23] deal with one component of the gradient of one component of the velocity field. More precisely, regularity of a weak solution $\mathbf{u}(t)$ of the Navier-Stokes equations is obtained under the assumption

Theorem 1.2 below establishes an extension of criterion (4) from the Navier-Stokes equations to the magneto-micropolar system (1).

Further regularity results for weak solutions of the Navier-Stokes equations are established in [1,4,5,6,7,10,25,26].

The main results of the current paper are:

Theorem 1.1. Let $(\mathbf{u}_{0}, \mathbf{w}_{0}, \mathbf{b}_{0}) \in L^2(\mathbb{R}^3)$ with $\nabla \cdot\mathbf{u}_{0},\nabla \cdot\mathbf{b}_{0} = 0.$ Let $T> 0$ and let

denote a weak solution of the magneto-micropolar equations (1) in $(0,T)$ satisfying the initial condition $(\mathbf{u},\mathbf{w},\mathbf{b})(0) = (\mathbf{u}_{0},\mathbf{w}_{0},\mathbf{b}_{0})$. If

then $(\mathbf{u},\mathbf{w},\mathbf{b})$ is $C^\infty$ in $\mathbb{R}^3\times(0,T)$ and the solution can be extended as a $C^\infty$ function beyond $T$.

Theorem 1.2. Let $(\mathbf{u}_{0}, \mathbf{w}_{0}, \mathbf{b}_{0}) \in L^2(\mathbb{R}^3)$ with $\nabla \cdot\mathbf{u}_{0},\nabla \cdot\mathbf{b}_{0} = 0.$ Let $T> 0$ and let

denote a weak solution of the magneto-micropolar equations (1) in $(0,T)$ satisfying the initial condition $(\mathbf{u},\mathbf{w},\mathbf{b})(0) = (\mathbf{u}_{0},\mathbf{w}_{0},\mathbf{b}_{0})$. If

then $(\mathbf{u},\mathbf{w},\mathbf{b})$ is $C^\infty$ in $\mathbb{R}^3\times(0,T)$ and the solution can be extended as a $C^\infty$ function beyond $T$.

An outline of the paper follows: There are two sections after the Introduction. In Section 2, we list definitions and notations used throughout the paper and recall results that play an important role in our proofs of the main results. Section 3 presents the proofs of Theorems $1.1$ and 1.2.

2.   Preliminaries

We introduce notations and definitions used in the paper.

● Boldface letters denote vector fields; for example,

● The Euclidean norm of any vector $\mathbf{a} = (a_{1},...,a_{n})\in \mathbb{R}^n$ is denoted and given by $|\mathbf{a}| = \sqrt{\sum_{i = 1}^n a_{i}^2}$.

● The notation $L^{\alpha}$ is used for the standard Lebesgue space equipped with the norm $\|\cdot\|_{\alpha}$, where $1\leq \alpha\leq \infty;$ more specifically,

and

where $\mathbf{a}:\mathbb{R}^3\rightarrow \mathbb{R}^n$ ($n\in\mathbb{N}$) is a mensurable function. We define the $L^2$-inner product of two vector functions by

where $\mathbf{c}\cdot \mathbf{d}: = \sum_{i = 1}^{n}c_id_i$ for $\mathbf{c} = (c_{1},...,c_{n}),\mathbf{d} = (d_{1},...,d_{n})\in \mathbb{R}^n$; and $\mathbf{a},\mathbf{b}:\mathbb{R}^3\rightarrow \mathbb{R}^n$ ($n\in\mathbb{N}$) are mensurable functions.

● Let $\nabla \mathbf{a} = (\nabla a_{1},...,\nabla a_{n} )$ denote the gradient of $\mathbf{a} = (a_{1},...,a_{n})\in \mathbb{R}^n$, where $\nabla a_{j} = (\partial_{1} a_{j},\partial_{2}a_{j},\partial_{3}a_{j})$, with $\partial_{i} = \partial/\partial x_{i}$ for all $i = 1,2,3$ and $j = 1,...,n.$

● The horizontal gradient is denoted by $\nabla_{h} \mathbf{a} = (\nabla_{h} a_{1},...,\nabla_{h} a_{n} ),$ where $\mathbf{a} = (a_{1},...,a_{n})\in \mathbb{R}^n$ and $\nabla_{h} a_{j} = (\partial_{1} a_{j},\partial_{2}a_{j})$, with $j = 1,...,n.$

● Here $\mathbf{a}\cdot \nabla : = \sum_{i = 1}^3 a_{i}\partial_{i}$, where $\mathbf{a} = (a_{1},a_{2},a_{3})\in \mathbb{R}^3$.

● $\nabla \times \mathbf{a}$ denotes the curl of the vector field $\mathbf{a} = (a_{1},a_{2},a_{3})\in \mathbb{R}^3$.

● Denote $\nabla \cdot \mathbf{a} = \sum_{i = 1}^3 \partial_ia_{i}$, where $\mathbf{a} = (a_{1},a_{2},a_{3})\in \mathbb{R}^3$.

● $\Delta$ represents the standard Laplacian operator.

● The horizontal Laplacian is denoted by $\Delta_{h} \mathbf{a} = (\Delta_{h} a_{1},...,\Delta_{h} a_{n} ),$ where $\mathbf{a} = (a_{1},...,a_{n})\in \mathbb{R}^n$ and $\Delta_{h} a_{j} = \sum_{i = 1}^{2} \partial_{i}^2 a_{j}$, with $j = 1,...,n.$

● Let ($X,\|\cdot\|$) be a Banach space and assume that $1\leq \beta\leq \infty$, $c,d\in \mathbb{R}, c<d.$ We denote by $L^{\beta}(c,d;X)$ the space of all measurable functions $f:(c,d)\rightarrow X$ with $\|f(\cdot)\|\in L^{\beta}(c,d)$ endowed with the norm $\|f\|_{L^{\beta}(c,d;X)}: = \left(\int_{c}^{d}\|f(t)\|^\beta\,dt\right)^{\frac{1}{\beta}}$, where $\beta<\infty$, and also $\|f\|_{L^{\infty}(c,d;X)} = \hbox{ess sup}_{c<t<d}\{\|f(t)\|\}$.

● We define a weak solution of (1) as follows: Let $T>0$ and $(\mathbf{u}_{0},\mathbf{w}_{0},\mathbf{b}_{0})\in L^{2}(\mathbb{R}^3)$, with $\nabla\cdot\mathbf{u}_{0} = \nabla\cdot\mathbf{b}_{0} = 0$. A mensurable function $(\mathbf{u},\mathbf{w},\mathbf{b})(t)$ is called a weak solution for (1) on $[0,T)$ if the following conditions hold

1. $(\mathbf{u},\mathbf{w},\mathbf{b})(t)\in L^{\infty}(0,T;L^2(\mathbb{R}^3))\cap L^{2}(0,T;H^1(\mathbb{R}^3))$;

2. the system (1) is satisfied in the sense of distributions;

3. the energy inequality holds, i.e.,

for all $0\leq t<T$.

● For brevity, dependencies on the variables $x$ and $t$ are often suppressed in our notation. For example, the function $t \to \| \mathbf{u}(\cdot,t)\|_{2}$ may also be written as $\| \mathbf{u}(t)\|_{2}$ or $\| \mathbf{u}\|_{2}$. Furthermore, as usual, the value of constants may change line by line in the paper.

Now, we enunciate the lemmas that will be applied in the proofs of our main results. The first one is proved in [16].

Lemma 2.1 (see [16]). Let $i,j,k$ be a permutation of $1,2,3$. Assume that

Then,

The second one was established in [23].

Lemma 2.2 (see [23]). Let $f\in L^6(\mathbb{R}^3)$ and $g\in L^2(\mathbb{R}^3)$ with $\partial_3 f\in L^2(\mathbb{R}^3)$ and $\nabla_h g\in L^2(\mathbb{R}^3)$. Then,

The third one was written in [2].

Lemma 2.3 (see [2]). Assume that $\theta, \lambda, \vartheta \in \mathbb{R}$ satisfy

Consider that $f \in H^1(\mathbb{R}^3),\, \partial_1 f, \partial_2 f \in L^{\lambda}(\mathbb{R}^3)$ and $\partial_3 f \in L^{\theta}(\mathbb{R}^3)$. Then, there exists a positive constant $C$ such that

In particular, if $\lambda = 2$ and $f \in H^1(\mathbb{R}^3)$, $\partial_3 f \in L^{\theta}(\mathbb{R}^3)$ (with $1 \leq \theta < \infty$), then there is a positive constant $C$ such that

3.   Proof of the main results

In this section we prove Theorems 1.1 and 1.2. In both results, it is necessary to consider $\epsilon \in(0,T)$ arbitrary in order to obtain $\delta\in (0,\epsilon)$ such that $(\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})(\delta)\in L^2(\mathbb{R}^3).$ It is known that there is a unique strong solution $(\mathbf{u}',\mathbf{w}',\mathbf{b}')\in C([\delta,T^*);H^1(\mathbb{R}^3))\cap L^2(\delta,T^*;H^2(\mathbb{R}^3))$ for the system (1) (see [9,11] and references therein) that satisfies $(\mathbf{u}',\mathbf{w}',\mathbf{b}')(\delta) = (\mathbf{u},\mathbf{w},\mathbf{b})(\delta)$, where $t = T^*$ is the maximum time of existence for this solution, and $(\mathbf{u}',\mathbf{w}',\mathbf{b}')\in C^{\infty}(\mathbb{R}^3\times(0,T^*))$ (since $\epsilon>0$ is arbitrary). Therefore, if $T< T^*$, one concludes that $(\mathbf{u},\mathbf{w},\mathbf{b})(t)$ is regular in $\mathbb{R}^3\times (0,T).$ On the other hand, assuming $T^* \leq T$, we will prove below that $\|(\nabla\mathbf{u},\nabla\mathbf{w},\nabla\mathbf{b})(t)\|_2$ is uniformly bounded for $t\in[\delta,T^*)$ (in Theorem 1.2) and bounded as $t\nearrow T^*$ (in Theorem 1.1). However, this is not possible. In fact, this boundedness would imply that $(\mathbf{u}',\mathbf{w}',\mathbf{b}')(t)$ could be extended beyond $t = T^*$, which contradicts the definition of $T^*$.

3.1.   Proof of Theorem 1.1

First, notice that by applying the product $(\cdot,\Delta_h \mathbf{u})_{2}$ to the first equation of the magneto-micropolar system $(1)$ and using the fact that $\nabla \cdot\mathbf{u} = 0$, one gets

Similarly, from the second and third equations in $(1)$, we obtain

and also

By adding the results $(10)$, $(11)$ and $(12)$ one infers

Let us examine the terms on the right hand side of the above equality. We have

Similarly, we get

where we have used that $\mathbf{b}$ is divergence free. Hence,

Furthermore,

On the other hand, by analysing the last term above it is easy to prove that it is actually null. In fact, since $\nabla \cdot \mathbf{u} = 0$, one has

Therefore,

Similarly, we obtain

Also notice that, by applying Cauchy-Schwarz's inequality, one has

By [22], the following estimate holds:

Consequently, from $(13)$, we infer

By applying Lemma 2.1, one obtains

By using Young's inequality, it follows that

where $\alpha = \min\{\mu,\nu,\gamma\}.$ Now, by integrating over $[T^*-\tau ,t]$ the inequality above ($\tau$ will be chosen a posteriori), we obtain

In order to estimate the term $\|(\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf {b})(t)\|_{2}$ for all $t\in[ T^*-\tau, T^*),$ we define

and

where $T^*-\tau \leq t<T^*.$ First of all, let us establish a relationship between $I$ and $J$. By $(15)$, one has

where we have applied Young's inequality, (6) and (9). By using Young's inequality again, we get

or equivalently,

The inequality (18) is useful to prove that $\|(\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf {b})(t)\|_{2}$ is bounded in the interval $[ T^*-\tau, T^*).$ In order to establish this last statement, we start noting that the system $(1)$ lets us conclude the following:

and also

where we used the fact that $\nabla \cdot \mathbf{u} = 0$. Hence, by adding the three equalities above, one obtains

Let us examine all the terms on the right hand side of the equality above. We have

Here

Similarly,

By using that $\mathbf{u}$ is divergence free, one also has

Therefore, using the above estimates, the equality $(20)$ yields

Following the same process, we conclude that

It is important to point out that the technique applied to $(\mathbf{u}\cdot \nabla \mathbf{w}, \Delta \mathbf{w})_2$ may not be useful when we consider the terms $-(\mathbf{b}\cdot \nabla \mathbf{b}, \Delta \mathbf{u})_2$ and $-(\mathbf{b}\cdot \nabla \mathbf{u}, \Delta \mathbf{b})_2$ (such expressions were obtained in $(19)$). However, we can argue as follows.

Consequently,

By using $\nabla \cdot \mathbf{b} = 0,$ we have

Let us estimate each term $J_{i}(t)$ for $i = 1,2,3.$ Hence,

Similarly, one obtains

and, by applying $\nabla \cdot \mathbf{b} = 0,$ we have

Replacing, in $(21)$, the estimates obtained for $J_{i}(t)$, $i = 1,2,3$, we get

Furthermore, notice that

where we have applied Cauchy-Schwarz's inequality. At last, Y. Zhou and M. Pokorný [29] proved that

Therefore, $(19)$ reads

By using Lemma 2.1, one gets

where $\alpha = \min\{\mu, \gamma,\nu\}$. Now, by integrating over $[T^*-\tau ,s]$, $s\leq t$, the inequality above yields

where we applied the definition of $I(t)$ given in (16). By Hölder's inequality, we have

for all $s\leq t$. In the inequality above the definitions of $I(t)$ and $J(t)$ were applied (see (16) and (17)). It follows, by using (17), that

By using Young's inequality, we infer

Consequently, by applying (18), we obtain

From the energy inequality (9), one concludes that there exists $0<\tau\ll 1$ such that

Now, we can obtain the desired estimate for $\|(\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})(t)\|$ in $[T^*-\tau,T^*)$. In fact, by replacing the bound above in (24), we get

The definition (17) establishes the proof of Theorem 1.1.

3.2.   Proof of Theorem 1.2

In order to prove Theorem 1.2 let us examine all the terms on the right hand side of $(13)$ in an alternative way. We have

since $\nabla \cdot \mathbf{u} = 0.$ By arguing in the same way, one gets

Notice that

and also

The reader might check that (14) assures the following estimate:

At last, Y. Zhou and M. Pokorný [29] proved that

By replacing all these last results obtained above in (13) and by using Young's inequality, one has

where $\alpha = \min\{\mu,\gamma,\nu\}.$ Hence,

By Lemmas 2.2 and 2.3, and also by (8), we obtain

By Young's inequality, one concludes

By applying Gronwall's inequality, we get

for all $t\in [\delta, T^*)$. By energy inequality (9), we can guarantee the following estimate:

In order to prove that the term $\|(\nabla \mathbf{u},\nabla \mathbf{w},\nabla \mathbf{b})(t)\|_2$ is bounded in $[\delta,T^*)$, we recall that (23) and Hölder's inequality imply

where $\alpha = \min\{\mu,\gamma,\nu\}$. By using (25) and Gagliardo-Nirenberg's inequality, one has

for all $t\in [\delta,T^*)$. By Young's inequality, we infer

By Gronwall's inequality,

This completes the proof of Theorem 1.2.

Acknowledgments

The authors would like to thank reviewers for their precious suggestions.