The anisotropic integrability logarithmic regularity criterion to the 3D micropolar fluid equations

1.   Introduction

Let us consider the following Cauchy problem of the incompressible micropolar fluid equations in three-spatial dimensions:

where $u = u(x, t)\in \mathbb{R}^{3}$, $\omega = \omega (x, t)\in \mathbb{R}^{3}$ and $\pi = \pi \left(x, t\right)$ denote the unknown velocity vector field, the micro-rotational velocity and the unknown scalar pressure of the fluid at the point $\left(x, t\right) \in \mathbb{R}^{3}\times (0, T)$, respectively, while $u_{0}, \omega _{0}$ are given initial data with $\nabla \cdot u_{0} = 0$ in the sense of distributions.

Theory of micropolar fluid equations was first proposed by Eringen ^[11] in 1966, which have important applications in fluid mechanics and material sciences and which enables to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations for the viscous incompressible fluids, for example of animal blood, liquid crystals and delute aqueous polymer solutions, etc. (see ^[28,29,31]). If $\omega = 0$, then (1.4) reduces to be the well-known Navier-Stokes equations. Besides its physical applications, the Navier-Stokes equations are also mathematically significant. Since Leray ^[24] and Hopf ^[23] constructed the so-called well-known Leray-Hopf weak solution $u(x, t)$ of the incompressible Navier-Stokes equation for arbitrary $u_{0}(x)\in L^{2}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0}(x) = 0$ in last century, the problem on the uniqueness and regularity of the Leray-Hopf weak solutions is one of the most challenging problem of the mathematical community. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results were established (see e.g. ^{[12,13,14,22,33]} and references there in).

Due to the importance of both physics and mathematics, the question of smoothness and uniqueness of weak solutions to (1.4) is one of the most challenging problems in the theory of PDE's. Galdi and Rionero ^[19], Łukaszewicz ^[26] considered the existence of weak solutions of the micropolar fluid flows (1.4). While the existence of regular solutions is still open problem, there are many interesting sufficient conditions which guarantee that a given weak solution is smooth (see ^{[4,9,15,16,17,18,32]} and references there in). In particular, as for the pressure criterion, Dong et al. ^[10] (see also Yuan ^[31]) showed that the weak solution becomes regular if the pressure satisfies

or

where $L^{p, \infty }$ and $\overset{\cdot }{B}_{\infty, \infty }^{0}$ denote Lorents space and homogeneous Besov space. Later on, Jia et al. ^[20] extended and improved Serrin's regularity criterion to the largest critical Besov spaces as

where $\frac{2}{q}+\frac{3}{p} = 2+r$, $\, \ \frac{3}{2+r} < p < \infty$ and$\, \, {-1 } < r\leq 1$.

Besides, some interesting logarithmical pressure regularity criteria of micropolar fluid equations are studied. In particular, in ^[21], Jia et al. refined this question by establishing a regularity criterion in terms of the partial derivative of the pressure in the Lebesgue space. More precisely, they showed that if the partial derivative of the pressure $\partial _{3}\pi$ satisfies

then the weak solution $(u, \omega)$ becomes a regular solution on $(0, T]$. (see, for instance ^[1,2,3] and the more recent papers ^[5,6,7,8] and the references therein).

Here we would like to give an improvement of the anisotropic regularity criterion of (1.5). Before giving the main result, we recall the definition of weak solutions for micropolar fluid equations (1.4) (see ^[26,27]).

Definition 1.1 (weak solutions). Let $(u_{0}, \omega _{0})\in L^{2}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0} = 0$ in the sense of distribution and $T > 0$. A measurable function $\left(u(x, t), \omega (x, t)\right)$ on $\mathbb{R}^{3}\times (0, T)$ is called a weak solution of (1.4) on $[0, T)$ if $(u, \omega)$ satisfies the following properties:

$\bf{(i)}$ $(u, \omega)\in L^{\infty }\left(\left(0, T\right); L^{2}(\mathbb{R}^{3})\right) \cap L^{2}\left(\left(0, T\right); H^{1}(\mathbb{R} ^{3})\right);$

$\bf{(ii)}$ $\nabla \cdot u = 0$ in the sense of distribution;

$\bf{(iii)}$ $(u, \omega)$ verifies (1.4) in the sense of distribution.

$\bf{(iv)}$ $(u, \omega)$ satisfies the energy inequality, that is,

We endow the usual Lebesgue space $L^{p}(\mathbb{R}^{3})$ with the norm $\left\Vert \cdot \right\Vert _{L^{p}}$. We denote by $\partial _{i} = \frac{ \partial }{\partial x_{i}}$ the partial derivative in the $x_{i}-$direction. Recall that the anisotropic Lebesgue space consists on all the total measurable real valued functions $h = h(x_{1}, x_{2}, x_{3})$ with finite norm

where $(i, j, k)$ belongs to the permutation group $S =$span$\left\{ 1, 2, 3\right\}$. Our main result is as follows:

Theorem 1.2. Let $(u_{0}, \omega _{0})\in L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0} = 0$ in $\mathbb{R}^{3}$. Suppose that $(u, \omega)$ is a weak solution of (1.4) in $(0, T)$. If the pressure satisfies the condition

where

then the weak solution $(u, \omega)$ becomes a regular solution on $(0, T]$.

This allows us to obtain the regularity criterion of weak solutions via only one directional derivative of the pressure. This extends and improve some known regularity criterion of weak solutions in term of one directional derivative, including the notable works of Jia et al. ^[21].

Remark 1.1. Criterion (1.6) can be viewed as a generalization of the recent result (1.5) of Jia-Zhang-Dong in ^[21]. Moreover, thanks to the fact that micropolar fluid equations (1.4) with $\omega = 0$ reduce to the 3D Navier-Stokes equations, we notice that our criterion (1.6) becomes the recent result of Liu-Dai ^[25] for the Navier-Stokes equations.

As an application of Theorem 1.5, we also obtain the following regularity criterion of weak solutions.

Corollary 1.3. Let $(u_{0}, \omega _{0})\in L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0} = 0$ in the sense of distributions. Assume that $(u, \omega)$ is a weak solution of (1.4) in $(0, T)$. If the pressure satisfies the condition

where

then the weak solution $(u, \omega)$ becomes a regular solution on $(0, T]$.

Let us consider the following Cauchy problem of the incompressible micropolar fluid equations in three-spatial dimensions:

where $u = u(x, t)\in \mathbb{R}^{3}$, $\omega = \omega (x, t)\in \mathbb{R}^{3}$ and $\pi = \pi \left(x, t\right)$ denote the unknown velocity vector field, the micro-rotational velocity and the unknown scalar pressure of the fluid at the point $\left(x, t\right) \in \mathbb{R}^{3}\times (0, T)$, respectively, while $u_{0}, \omega _{0}$ are given initial data with $\nabla \cdot u_{0} = 0$ in the sense of distributions.

Theory of micropolar fluid equations was first proposed by Eringen ^[11] in 1966, which have important applications in fluid mechanics and material sciences and which enables to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations for the viscous incompressible fluids, for example of animal blood, liquid crystals and delute aqueous polymer solutions, etc. (see ^[28,29,31]). If $\omega = 0$, then (1.4) reduces to be the well-known Navier-Stokes equations. Besides its physical applications, the Navier-Stokes equations are also mathematically significant. Since Leray ^[24] and Hopf ^[23] constructed the so-called well-known Leray-Hopf weak solution $u(x, t)$ of the incompressible Navier-Stokes equation for arbitrary $u_{0}(x)\in L^{2}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0}(x) = 0$ in last century, the problem on the uniqueness and regularity of the Leray-Hopf weak solutions is one of the most challenging problem of the mathematical community. Later on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results were established (see e.g. ^{[12,13,14,22,33]} and references there in).

Due to the importance of both physics and mathematics, the question of smoothness and uniqueness of weak solutions to (1.4) is one of the most challenging problems in the theory of PDE's. Galdi and Rionero ^[19], Łukaszewicz ^[26] considered the existence of weak solutions of the micropolar fluid flows (1.4). While the existence of regular solutions is still open problem, there are many interesting sufficient conditions which guarantee that a given weak solution is smooth (see ^{[4,9,15,16,17,18,32]} and references there in). In particular, as for the pressure criterion, Dong et al. ^[10] (see also Yuan ^[31]) showed that the weak solution becomes regular if the pressure satisfies

or

where $L^{p, \infty }$ and $\overset{\cdot }{B}_{\infty, \infty }^{0}$ denote Lorents space and homogeneous Besov space. Later on, Jia et al. ^[20] extended and improved Serrin's regularity criterion to the largest critical Besov spaces as

where $\frac{2}{q}+\frac{3}{p} = 2+r$, $\, \ \frac{3}{2+r} < p < \infty$ and$\, \, {-1 } < r\leq 1$.

Besides, some interesting logarithmical pressure regularity criteria of micropolar fluid equations are studied. In particular, in ^[21], Jia et al. refined this question by establishing a regularity criterion in terms of the partial derivative of the pressure in the Lebesgue space. More precisely, they showed that if the partial derivative of the pressure $\partial _{3}\pi$ satisfies

then the weak solution $(u, \omega)$ becomes a regular solution on $(0, T]$.

Here we would like to give an improvement of the anisotropic regularity criterion of (1.5). Before giving the main result, we recall the definition of weak solutions for micropolar fluid equations (1.4) (see ^[26]).

Definition 1.4 (weak solutions). Let $(u_{0}, \omega _{0})\in L^{2}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0} = 0$ in the sense of distribution and $T > 0$. A measurable function $\left(u(x, t), \omega (x, t)\right)$ on $\mathbb{R}^{3}\times (0, T)$ is called a weak solution of (1.4) on $[0, T)$ if $(u, \omega)$ satisfies the following properties:

$\bf{(i)}$ $(u, \omega)\in L^{\infty }\left(\left(0, T\right); L^{2}(\mathbb{R}^{3})\right) \cap L^{2}\left(\left(0, T\right); H^{1}(\mathbb{R} ^{3})\right);$

$\bf{(ii)}$ $\nabla \cdot u = 0$ in the sense of distribution;

$\bf{(iii)}$ $(u, \omega)$ verifies (1.4) in the sense of distribution.

$\bf{(iv)}$ $(u, \omega)$ satisfies the energy inequality, that is,

We endow the usual Lebesgue space $L^{p}(\mathbb{R}^{3})$ with the norm $\left\Vert \cdot \right\Vert _{L^{p}}$. We denote by $\partial _{i} = \frac{ \partial }{\partial x_{i}}$ the partial derivative in the $x_{i}-$direction. Recall that the anisotropic Lebesgue space consists on all the total measurable real valued functions $h = h(x_{1}, x_{2}, x_{3})$ with finite norm

where $(i, j, k)$ belongs to the permutation group $S =$span$\left\{ 1, 2, 3\right\}$. Our main result is as follows:

Theorem 1.5. Let $(u_{0}, \omega _{0})\in L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0} = 0$ in $\mathbb{R}^{3}$. Suppose that $(u, \omega)$ is a weak solution of (1.4) in $(0, T)$. If the pressure satisfies the condition

where

then the weak solution $(u, \omega)$ becomes a regular solution on $(0, T]$.

This allows us to obtain the regularity criterion of weak solutions via only one directional derivative of the pressure. This extends and improve some known regularity criterion of weak solutions in term of one directional derivative, including the notable works of Jia et al. ^[21].

Remark 1.2. Criterion (1.6) can be viewed as a generalization of the recent result (1.5) of Jia-Zhang-Dong in ^[21]. Moreover, thanks to the fact that micropolar fluid equations (1.4) with $\omega = 0$ reduce to the 3D Navier-Stokes equations, we notice that our criterion (1.6) becomes the recent result of Liu-Dai ^[25] for the Navier-Stokes equations.

As an application of Theorem 1.5, we also obtain the following regularity criterion of weak solutions.

Corollary 1.6. Let $(u_{0}, \omega _{0})\in L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0} = 0$ in the sense of distributions. Assume that $(u, \omega)$ is a weak solution of (1.4) in $(0, T)$. If the pressure satisfies the condition

where

then the weak solution $(u, \omega)$ becomes a regular solution on $(0, T]$.

2.   Preliminaries

Before to prove our main result, we first recall the following result proved in ^[30].

Lemma 2.1. Let $n\geq 2$ be a natural number, $\gamma _{i}, t_{i}\geq 0$, $p_{i}\in (1, +\infty)$, $i = 1, 2, 3, ..., n$ and suppose that

Then there exists $C > 0$ such that for every $f\in C^{\infty }\cap L^{2}$

Next, we recall the following Gagliardo-Nirenberg interpolation inequality in $\mathbb{R}^{1}$.

Lemma 2.2. Let $1\leq \kappa, \mu, \nu < \infty$ satisfy

Assume that $\varphi \in H^{1}(\mathbb{R}^{3})$. Then there exists a constant $C > 0$ such that

The crucial tool in this paper is the following result, which plays important role in proving our main result.

Lemma 2.3. Let $r > 1$ and $1 < \gamma \leq \alpha < \infty$. Then for $f, g, \varphi \in C_{0}^{\infty }(\mathbb{R}^{3})$, we have

where $0\leq \theta \leq 1$ satisfying

and

and $C$ is a constant independent of $f, g, \varphi$.

Invoking Hölder's inequality and Fubini's theorem, we obtain

Notice that

Moreover, by Hölder's inequality and (2.1), we obtain

where we use the interpolation theorem

and

We can use Lemma 2.1, where we put $n = 2$, $\delta = \frac{2r}{r-1}$, $p_{1} = p_{2} = \frac{2}{r}$, $\gamma _{1} = \gamma _{1} = \frac{1}{2}$, $t_{1} = t_{2} = r-1$ and estimate by applying Hölder's inequality

So by applying Minkowski's inequality, we obtain

Similarly, we have

Thus,

We recall the following result according to Dong et al. ^[10], that will be used in the proof of Theorem 1.5.

Lemma 2.4. Suppose $(u_{0}, \omega _{0})\in L^{s}(\mathbb{R}^{3})$, $s > 3$ with $\nabla \cdot u_{0} = 0$ in $\mathbb{R}^{3}$. Then there exists $T > 0$ and a unique strong solution $(u, \omega)$ of the 3D micropolar fluid equations (1.4) such that

Moreover, let $(0, T_{0})$ be the maximal interval such that $(u, \omega)$ solves (1.4) in $C\left((0, T_{0}); L^{s}(\mathbb{R}^{3})\right)$, $s > 3$. Then, for any $t\in (0, T_{0}),$

with the constant $C$ independent of $T_{0}$ and $s$.

By a strong solution we mean a weak solution $(u, \omega)$ such that

It is well-known that strong solution are regular (say, classical) and unique in the class of weak solutions.

3.   Proof of Theorem 1.5

This section is devoted to the proof of Theorem 1.5.

Proof: First, we multiply both sides of the equation (1.4)$_{1}$ by $u\left\vert u\right\vert ^{2}$, and integrate over $\mathbb{R}^{3}$. After suitable integration by parts, we obtain

where we used the following identities due to divergence free condition:

Note that

Multiplying the second equation of (1.4) by $\omega \left\vert \omega \right\vert ^{2}$, then integrating the resulting equation with respect to $x$ over $\mathbb{R}^{3}$ and using integrating by parts, we obtain

where we have used the fact that $\nabla$div$\omega = \nabla \times (\nabla \times \omega)+\Delta \omega$ yields

Combining (3.1) and (3.2) together, it follows that

With the use of Hölder's inequality and Young's inequality, the first two terms on the right-hand side of (3.3) is bounded by

Let us now estimate the integral $A_{1}$. The Cauchy inequality implies that

Let us estimate the integral

on the right-hand side of (3.5). Before turning to estimate $I$, it is well-known that for the micropolar fluid equations in $\mathbb{R}^{3}$, we have the following relationship between $\pi$ and $u$ and Calderón-Zygmund inequality

We select that $a = \frac{\alpha (\gamma +\alpha \gamma -\alpha)}{\alpha -\gamma }$ and $b = \frac{\gamma +\alpha \gamma -\alpha }{\alpha (\gamma -1)}$ in Lemma 2.3, then the selected $a$ and $b$ satisfy (2.4). Then we can estimate $I$ as follows

where $\alpha, \beta, r$ and $\theta$ satisfy the following identities

Using the fact that $2\leq \lambda < 3$, we choose $r = \frac{(4-\lambda)\alpha \gamma }{\gamma +\alpha \gamma -\alpha }$, then it follows from (3.6)$_{3}$ that $\beta = \frac{(3-\lambda)\gamma }{\gamma -1}$. Now, on the one hand, observe that

On the other hand, since

and since $\alpha \geq \gamma$, we get

But you know, $\lambda$ must be less than $3$, hence

which implies that $(3-\lambda)\alpha \gamma +(\alpha -\gamma) > 0$. Gathering these estimates together, we obtain

and it is clear that

Now using Hölder inequality with exponents $\frac{\lambda \alpha \gamma -\alpha -2\gamma }{2[(3-\lambda)\alpha \gamma -\gamma +\alpha]}$ and $\frac{2(3-\lambda)\alpha \gamma -\alpha (\lambda \gamma -3)}{2[(3-\lambda)\alpha \gamma -\gamma +\alpha]}$, $I$ can be further estimated as

when $\frac{\lambda \alpha \gamma -\alpha -2\gamma }{2[(3-\lambda)\alpha \gamma -\gamma +\alpha]} = 0$ $($i.e. $\alpha = \gamma = \frac{3}{\lambda })$ or

when $0 < \frac{\lambda \alpha \gamma -\alpha -2\gamma }{2[(3-\lambda)\alpha \gamma -\gamma +\alpha]} < 1$ $($i.e. $\frac{3}{\lambda } < \gamma \leq \alpha < \frac{1}{\lambda -2})$ and $\beta = \frac{(3-\lambda)\gamma }{\gamma -1}$.

Combining all the estimates from above, we get

Defining

and thanks to

inequality (3.7) implyes that

and hence

Thanks to $(u, \omega)$ is a weak solution of the 3D micropolar equations (1.4), that is

together with the interpolation inequality yields that

On the other hand, since

and

it is easy to see that

and consequently

Hence, one has

Applying the Gronwall inequality yields that

Now, we are in a position to complete the proof of Theorem 1.5. From Lemma 2.4, it follows that there exists $T_{0} > 0$ and the smooth solution $(\widetilde{u}, \widetilde{\omega })$ of (1.4) satisfying

Since the weak solution $(u, \omega)$ satisfies the energy inequality, we may apply Serrin's uniqueness criterion to conclude that

Thus, it is sufficient to show that $T_{0} = T$. Suppose that $T_{0} < T$. Without loss of generality, we may assume that $T_{0}$ is the maximal existence time for $(\widetilde{u}, \widetilde{\omega })(t)$. By lemma 2.4 again, we find that

On the other hand, from (3.10), we know that

which contradicts with (3.11). Thus, $T_{0} = T$. This completes the proof of Theorem 1.5.

Acknowledgments

This work was done while the second author was visiting the Catania University in Italy. He would like to thank the hospitality and support of the University, where this work was completed. This research is partially supported by PRIN 2019 The fourth author wish to thank the support of "RUDN University Program 5-100".

Conflict of interest

All authors declare no conflicts of interest in this paper.