Regularity and uniqueness of 3D compressible magneto-micropolar fluids

1.   Introduction

This paper focuses on the three-dimensional compressible magneto-micropolar fluid system in $\mathbb{R}^3$ ^[17,18,24]:

Here, $\rho\geqslant 0$ is the density, $\mathbf{u}$ is the velocity, $\mathbf{w}$ is the micro-rotational velocity, $\mathbf{b}$ is the magnetic field and $P(\rho) = A\rho^\gamma\; (A > 0, \; \gamma > 1)$ is the pressure. The parameters $\eta, \kappa, \beta, \eta^{\prime}, \kappa^{\prime}$ and $\alpha$ are constants satisfying

For the completeness of equations (1.1), we supplement the conditions

and

When we only consider the effects of the velocity field and the magnetic field in (1.1) (i.e., $\mathbf{ w} = 0, \beta = 0$), the system (1.1) becomes the compressible magnetohydrodynamic (MHD) system, which has been widely discussed; see ^{[4,5,9,11,12,25,26]} and references therein. When we lose sight of the effects of magnetic fields, the system (1.1) becomes a micropolar model whose theory was introduced by Eringen ^[10] and Lukaszewicz ^[16]. Chen-Xu-Zhang in ^[6] proved the existence of global weak and smooth solutions in cases of small energy. In bounded or unbounded domain $\Omega \subset \mathbb{R}^3$, Chen ^[8] also got the local existence of strong solutions in the context of large initial data.

For system (1.1), there are still many papers discussing this model ^[3,7,20]. Wei, Guo, and Li ^[18] found the global existence and decay properties of smooth solutions with small initial perturbation ^[23]. Later, Tong and Tan ^[17] improved the results of ^[18], and obtained the same decay property between the linearized equations and nonlinear system. Later, Xu, Tan, and Wang in ^[21] considered the system in a bounded region and derived the global well-posedness strong solutions. Zhang-Cai in ^[24] considered the system (1.1) in a periodic region and derived the well-posedness of solutions. Moreover, in the context of small conditions and symmetrical external forces, the periodic solution was attained. Utilizing Hoff's method ^[13], Xu, Tan, and Wang in ^[22] derived the global weak solutions in context of discontinuous initial data. Very recently, Xu and Zhong in ^[20] proved the local well-posedness of strong solutions to (1.1) with vacuum. Chen, Sun, and Zhong in ^[3] deduced the well-posedness of global classical solutions in case of vacuum states. However, the results obtained in ^[6,7,8] still require some compatibility conditions

for some $(g_1, g_2)\in L^2$. In case of density including vacuum, it is known from (1.4) that

where, $\mathbf{ \dot{g}} = \mathbf{ g}_t + \mathbf{u} \cdot \nabla \mathbf{ g}$ stands for the material derivative. If $\rho_0 > 0$, then (1.4) becomes

To make sure (1.4)/(1.5) or (1.6) are true, the initial velocity and micro-rotational velocity satisfy

Thus,

where, $\|(\mathbf{g}, \mathbf{ h})\|_{L^p}\triangleq \|\mathbf{ g}\|_{L^p}+\|\mathbf{ h}\|_{L^p}$. Higher-order estimates can be deduced from (1.8).

Similar to ^[13,15], we set

where, the quantities $F_1$ and $F_2$ represent the "effective viscous flux", and $G_1$ and $G_2$ stand for the vorticity ^[6]. One can deduce from (1.9) that

Our purpose in this article is to modify the method of ^[19] and set up the following theorem.

Theorem 1.1. For $s\in [9/2, 6)$, assume that $(\rho_0, \mathbf{ u}_0, \mathbf{ w}_0, \mathbf{ b}_0)$ satisfies

There exists a constant $\varepsilon > 0$, depending on $\eta, \kappa, \beta, \eta^{\prime}, \kappa^{\prime}, \gamma, \alpha, A, \inf \rho_0$, $\sup \rho_0$, $\|\nabla \mathbf{ u}_0\|_{L^2}$, $\|\nabla \mathbf{ w}_0\|_{L^2}$, and $\|\nabla \mathbf{ b}_0\|_{L^2}$, such that if

the systems (1.1)–(1.3) possess a global uniqueness solution $(\rho, \mathbf{ u}, \mathbf{ w}, \mathbf{ b})$ in $\mathbb{R}^3\times (0, \infty)$ satisfying

with $1 < \ell < (4s)/(5s-6)$.

2.   Preliminaries

In this section, we begin to hammer at the derivations to obtain the global a priori estimates. Let $(\rho, \mathbf{ u}, \mathbf{ b}, \mathbf{ w})$ stand for a smooth solution of (1.1)–(1.4) on $\mathbb{R}^3 \times [0, T]$ for some $0 < T < \infty$, then the system (1.1) becomes

Thus, we have the following lemma, which can be found in ^[6,15].

Lemma 2.1. Given positive numbers $N$ (not necessarily small) and $\hat{\rho} > 2$, assume that $(\rho_0, \mathbf{ u}_0$, $\mathbf{ w}_0, \mathbf{ b}_0)$ satisfies

then there exist constants $L > 0$ and $\varepsilon > 0$, depending on $\eta, \kappa, \beta, \eta^{\prime}, \kappa^{\prime}, \gamma, N, \alpha, A, \; and \; \hat{\rho}$, such that if

where $J(\cdot)$ stands for the potential energy density

then

Remark 2.1. The estimations given by (2.4)–(2.7) do not depend on $T$ and $\inf \rho_0$. In addition, if $\inf \rho_0 > 0$, then $J(\cdot)$ implies

Remark 2.2. We infer from (2.7) that

Lemma 2.2. Under the circumstance of (2.2) and (2.3), then there exists positive constant $\varepsilon_1$, such that

provided $\mathcal{S}_0\leqslant \varepsilon_1$.

Proof. First, operate $t \dot{u}^j [\tfrac{d}{dt}+ {\text{div}}(\mathbf{u}\cdot)]$ and $t \dot{w}^j [\tfrac{d}{dt}+ {\text{div}}(\mathbf{u}\cdot)]$ to (2.1)$_2^j$ and (2.1)$_3^j$, respectively. Integration by parts gives

Based upon the integration by parts, one has

and

Similarly

and

Taking notice of (1.1)$_1$

The inequality (2.4), together with the integration by parts, yields

Next, for $I_6$ and $I_7$, we see that

Next, for $I_9$, the integration by parts gives

where, we have used the Gagliardo-Nirenberg's inequality:

Integration by parts, together with (2.11), yields

Substituting (2.12)–(2.15), (2.17)–(2.20), and (2.22) into (2.11), we obtain

provided

Taking note of $\|\nabla \mathbf{ b}_t\|_{L^2}$, then

Thanks to $\mathbf{u}_t = \dot{ \mathbf{u}}- \mathbf{u} \cdot \nabla \mathbf{u}$, we get

We utilize (2.21) to get that

and

Putting (2.25)–(2.27) into (2.24), one has

The inequalities (2.23), (2.28), (2.5), and (2.6) yield

We deduce from (1.9) and (1.10) that

which can be found in ^[3]. The inequalities (2.4)–(2.8), (2.21), and (2.30) give

The inequality (2.1)$_4$ gives

thus

Based upon (2.1)$_4$, it is easy to get that

thus

Therefore, putting (2.31)–(2.33) into (2.29), we obtain (2.10). □

Lemma 2.3. Under the circumstance of (2.2) and (2.3),

where $(r, s)$ satisfies

Proof. The inequality (2.4) together with (2.21) leads to

thus, by using Hölder's inequality and the inequalities (2.6) and (2.10) gives

Thanks to $r\in (3, 6)$ and $1 < s < \frac{4r}{5r-6} < 2$, one can deduce

With the help of (1.9) and (1.10), we get

and for any $m \in [2, 6]$,

thus, we have from (2.4)–(2.7), (2.37), (2.30), (2.38) that

Due to $s\in (1, 4/3)$, one has

which, together with (2.36) and (2.39), gives (2.34). □

In the next step, we mainly focus on estimating that $\|\nabla \mathbf{ u} \|_{L^p}$ holds for $p\in (1, \infty)$ and $p = \infty$, respectively. Due to $-\Delta \mathbf{ u} = \nabla {\rm div}\mathbf{ u}-\nabla \times {\rm rot}\mathbf{ u}$, for $0 < p < \infty$, $\|\nabla \mathbf{ u} \|_{L^p}\leqslant C\left(\|{\rm div} \mathbf{ u}\|_{L^p}+\| {\rm rot} \mathbf{ u}\|_{L^p}\right)$. However, for $p = \infty$, we have the following lemma.

Lemma 2.4. (Beale-Kato-Majda type inequality (cf. ^[1,14])) For $k \in \mathbb{Z}^+$ and $s\in(1, +\infty)$, let $D^{k, s}\triangleq \Big\{\mathbf{ g}\in L^1_{loc} | \partial^k \mathbf{ g}\in L^s\}$ and $D^{1}\triangleq D^{1, 2}$ be the homogeneous Sobolev spaces. For $\mathbf{ g}\in D^1\cap D^{2, s}$ with $s \in (3, +\infty)$, there exists a positive constant $C(s) > 0$, such that for all $\nabla \mathbf{ g} \in L^2 \cap D^{1, s}$,

Lemma 2.5. Let (1.11) and (1.12) be in force, then

where $r\in (3, 6)$ and $s\in (1, \infty)$ are the same ones as in (2.35).

Proof. We operate $\nabla$ to both sides of (2.1)$_1$ and multiply the result equation by $|\nabla \rho|^{r-2}\nabla\rho$ for $r \in [2, 6]$. One can deduce from (2.4) and integration by parts that

Recalling that $\mathcal{G}\triangleq-(\eta+\beta)\Delta-(\eta+\kappa-\beta)\nabla {\text{div}}$ is a strong elliptic operator (^[2]), the inequalities (2.1)$_2$ and (2.4) give

which, combined with (2.42), gives

The combination of (2.40) and (2.43) gives

for any $3 < r < 6$. Taking (2.45) into (2.44), one gets

where

Noting that the relationship $\ln(e+z) \leqslant (e+z)^\tau$ holds for any $z \geqslant 0$ and $\tau > 0$, the inequality (2.33) gives

which, combined with (2.46), leads to

The inequalities (2.7) and (2.34) give

for any $r, s$ being as the ones in (2.31). Next, taking $r = 2$ in (2.42), one has

thus

Note that the Eq (2.1)$_1$ gives

which, combined with (2.47)–(2.49), gives (2.41). □

Lemma 2.6. Assume that $\rho_0$ satisfies $\inf_{x \in \mathbb{R}^3} \rho_0(x) \geqslant \check{\rho} > 0$, then there exists a constant $c > 0$, depending on $\check{\rho}$ and $T$, such that

and

Proof. The inequality (2.34) gives

which yields (2.50).

Next, the inequalities (2.1)$_1$, (2.4), and (2.5) give

thus, using (2.4), (2.5), (2.10), (2.41), and (2.50), one has

Taking note of $\dot{ \mathbf{u}}$,

and the inequalities (2.4), (2.6), (2.10), (2.50), and (2.52) give

and

On the other hand, due to (2.1)$_3$, (2.4)–(2.6), and (2.9), one has

Utilizing (2.52)–(2.55), we can obtain the estimate (2.51). □

Based upon the foregoing, the following estimation of $\|\nabla \mathbf{ u}\|_{L^3}$ is very crucial for this article.

Lemma 2.7. Under the circumstance of Theorem 1.1, we have

Proof. Utilizing the inequality $\|\nabla \mathbf{ u}\|_{L^m}\leqslant C\| ({\rm div} \mathbf{ u}, {\rm rot} \mathbf{ u})\|_{L^m}$ for $1 < m < \infty$, thus, we just have to estimate $\| {\rm div} \mathbf{ u}\|_{L^m}$ and $\|{\rm rot} \mathbf{ u}\|_{L^m}$. Operating $div$ and $rot$ to (2.1)$_2$, one gets

and

We first to estimate $\|{\rm div} \mathbf{u}\|_{L^3}$. Multiplying (2.57) by $| {\text{div}} \mathbf{u}| {\text{div}} \mathbf{u}$, one can deduce from integration by parts that

With the help of (2.1)$_2$ and the integration by parts, we get

The inequality (2.6), together with (2.41) gives

where, we used the following facts:

In terms of $J_{1, 2}$ and the inequality (2.41), together with (2.62), gives

and

Putting (2.61), (2.63)–(2.65) into (2.60), one has

Next, by virtue of (2.4), (2.6), (2.41), and (2.62),

and

Noticing that ${\rm div}\mathbf{ b} = 0$, the inequality of (2.6) gives

Substituting (2.66)–(2.69) into (2.59), one has

Next, we estimate $\|{\rm rot} \mathbf{u}\|_{L^3}$. We first multiply (2.58) by $|{\rm rot} \mathbf{ u}| {\rm rot} \mathbf{ u}$ and then deduce from integration by parts that

The Hölder's inequality gives

Noting that

thus

By virtue of (2.6), (2.23), (2.41), (2.62), we get

and

For $I_{2, 3}$, we utilize the fact

and take $\phi = \ln\rho$, $\psi = {\rm div} \mathbf{ u}$, and $v = |{\rm rot} \mathbf{ u}|{\rm rot} \mathbf{ u}$, then

The inequality (2.41), together with (2.4), leads to

and

Taking (2.74)–(2.79) into (2.73), one has

Similarly,

and

We notice that ${\text{div}} \mathbf{b} = 0$, thus

Putting (2.72), (2.73) and (2.80)–(2.83) into (2.71), we obtain

We close the estimations. The inequality (2.70), together with (2.84), gives

Due to

thus

Thanks to $\|\nabla \mathbf{u}\|_{L^r}\leqslant C \left(\|{\rm div} \mathbf{u} \|_{L^r}+\|{\rm rot} \mathbf{u}\|_{L^r}\right)\; (\forall\; r > 1)$, we obtain from (2.6), (2.51), (2.85), and (2.86) that

which, allied with (2.51), gives (2.56). □

Lemma 2.8. Under the circumstance of Theorem 1.1, then

Proof. Operating $div$ and $rot$ to (2.1)$_3$, one has

and

We multiply (2.88) by $| {\text{div}} \mathbf{w}| {\text{div}} \mathbf{w}$ and integrate by parts over $\mathbb{R}^3$, and one has

The inequalitis (2.1)$_2$ and (2.60) give

Similar to the estimation of (2.67), we get

Substituting (2.91) and (2.92) into (2.90), we have

Multiplying (2.89) by $|{\rm rot} \mathbf{w}| {\rm rot} \mathbf{w}$ and integrating by parts over $\mathbb{R}^3$, we deduce

Hence

Similar to the estimation of (2.85), we can get (2.87) from (2.93) and (2.94). □

3.   Proof of Theorem 1.1

In this section, we mainly focus on proving in Theorem 1.1 holds, based on the global a priori estimates that have been obtained in Section 2. Actually, the global existence can be established by modifying the method in ^[6]. In the next step, by modifying the ideas of ^[19], we will prove the uniqueness of solutions holds.

Proof of uniqueness. Let $(\rho_1, \mathbf{u}_1, \mathbf{w}_1, \mathbf{b}_1)$ and $(\rho_2, \mathbf{u}_2, \mathbf{w}_2, \mathbf{b}_2)$ be two solutions to the system (2.1), (1.2) and (1.3) on $\mathbb{R}^3\times[0, T]$ that satisfy (1.13) and have the same initial data. Define

then it follows from (2.1)$_1$ that

Multiplying (3.1) by $\varphi$, one can deduce from integration by parts that

Due to (1.13), we know that $\| {\text{div}} \mathbf{u}_2\|_{L^\infty}\in L^1(0, T)$. Thus, the inequality (3.2) yields

Since it holds that $\dot{ \mathbf{u}}_2 = \mathbf{u}_{2t}+ \mathbf{u}_2 \cdot \nabla \mathbf{u}_2$, we get

Multiplying (3.4) by $\mathbf{v}$ and integrating by parts, we get

Thus, the inequality (1.13), together with (2.50) and (2.56), leads to

On the other hand, the inequality (2.1)$_3$ gives

Multiply (3.6) by $\varpi$ and integrate by parts, and we get

thus

Note that

Multiply (3.8) by $\phi$, and one can get form integration by parts that

We deduce from (2.56) that

and

We have from (1.13) that

Taking (3.10)–(3.13) into (3.9), one has

which, combined with (3.3), (3.5), and (3.7), gives

Let

then, the inequality (3.14) gives

where

We know from (1.13) that $\psi(t)\in L^1(0, T)$; thus, the inequality (3.15), combined with Gronwall's inequality, gives

so

which, combined with (3.3), gives

Thus, Theorem 1.1 is proved. □

4.   Conclusions

From the discussion in the previous sections, we have concluded that the three-dimensional compressible magneto-micropolar fluid system (1.1) possesses a global and unique solution in $\mathbb{R}^3$, as follows:

For $s\in [9/2, 6)$, assume that $(\rho_0, \mathbf{ u}_0, \mathbf{ w}_0, \mathbf{ b}_0)$ satisfies

There exists a constant $\varepsilon > 0$, depending on $\eta, \kappa, \beta, \eta^{\prime}, \kappa^{\prime}, \gamma, \alpha, A, \inf \rho_0$, $\sup \rho_0$, $\|\nabla \mathbf{ u}_0\|_{L^2}$, $\|\nabla \mathbf{ w}_0\|_{L^2}$, and $\|\nabla \mathbf{ b}_0\|_{L^2}$, such that if

the systems (1.1)–(1.3) possess a global uniqueness solution $(\rho, \mathbf{ u}, \mathbf{ w}, \mathbf{ b})$ in $\mathbb{R}^3\times (0, \infty)$ satisfying

with $1 < \ell < (4s)/(5s-6)$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We sincerely thank the Associate Editor and the anonymous referees for their carefully reading and helpful suggestions that led to the improvement of the paper.

M.Y. Zhang was partially supported by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2021QA049) and the Science and Technology Project of Weifang (2022GX006).

Conflict of interest

The author declares no conflict of interest.