On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity

1.   Introduction

The nonhomogeneous incompressible micropolar fluids with density-dependent viscosity (^[12,20]) in $\mathbb{R}^3$ read as follows:

where $\rho, {\bf{u}} = (u^1, u^2, u^3)$, ${\bf{w}} = (w^1, w^2, w^3)$, and $P$ denote the fluid density, velocity, micro-rotational velocity, and pressure, respectively. The viscosity coefficient $\mu(\rho)$ satisfies

for some positive constant $\alpha$, while the constants $\mu^{\prime}$ and $\lambda^{\prime}$ are the angular viscosities satisfying $\mu^{\prime} > 0$ and $\mu^{\prime}+\lambda^{\prime}\geqslant 0$, and the constant $\zeta > 0$ denotes the dynamic micro-rotation viscosity.

In this paper, we consider the Cauchy problem of (1.1)–(1.2) with the far-field behavior

and the initial conditions

A micropolar fluid system is the study of fluids that exhibit micro-rotational effects and micro-rotational inertia and can be viewed as non-Newtonian fluids. It can be used to describe many phenomena that appear in a large number of complex fluids, such as suspensions, animal blood, and liquid crystals. The micropolar fluid system reduces to the Navier-Stokes equations when there is no microstructure ($\zeta = 0$ and ${\mathbf w} = 0$) and has been discussed by many mathematicians (see ^{[1,2,5,6,7,8,9,10,13,17,19]}, and references therein).

When it comes to the case that $\zeta\neq 0$ and ${\mathbf w}\neq 0$, there have been substantial developments on the global regularity problem concerning nonhomogeneous micropolar fluids (1.1) with constant viscosity $\mu$. When the initial density is strictly away from vacuum, Braz e Silva and his cooperators ^[4] investigated the global existence and uniqueness of solutions for 3D nonhomogeneous asymmetric fluids by using an approach and Lagrangian coordinates under suitable initial conditions. Qian-Chen-Zhang ^[15] studied the global existence of weak and strong solutions to 3D nonhomogeneous incompressible asymmetric fluid equations. For initial velocities sufficiently small in the critical Besov space, global Fujita-Kato type solutions with initial density in the bounded function space and that have a positive lower bound are obtained, and this result extends the classical one on the life-span by Leray. Subsequently, Qian-He-Zhang ^[16] investigated the global existence and uniqueness of the solutions for the 2D inhomogeneous incompressible asymmetric fluids, with the initial (angular) velocity being located in sub-critical Sobolev spaces $H^s(\mathbb{R}^2) (0 < s < 1)$ and the initial density being bounded from above and below by some positive constants. In particular, the uniqueness of the solution in ^[16] is also obtained without any more regularity assumptions on the initial density. When the initial density contains a vacuum state, Braz e Silva and Santos ^[3] established the existence of global in-time weak solutions for the equations of asymmetric incompressible fluids with variable density. Zhang-Zhu ^[18] proved the global existence of strong solutions under the condition of the following compatibility:

for some $(P(\rho_0), g_1, g_2)\in H^1 \times L^2 \times L^2$.

When we consider the case of $\mu = \mu(\rho)$, Liu-Zhong ^[12] showed that the initial boundary value problem of 2D nonhomogeneous micropolar fluids with density-dependent viscosity has a global and unique strong solution under the assumption of the smallness of $\|\nabla \mu(\rho_0)\|_{L^q}$. Qian-Qu ^[14] investigated the 3D inhomogeneous incompressible asymmetric fluids system and proved local well-posedness for initial velocity in the critical Besov space $\dot{B}_{p, 1}^{3/p}$ for $1 < p < 6$ and initial density $\rho_0$ satisfying that $\rho_0-1$ is in the critical Besov space and that $\rho_0$ is bounded away from zero. Zhong ^[20] also considered the same model in 3D cases and established the global existence and uniqueness of strong solutions, provided that the initial energy is sufficiently small. It is worth noting that there is no need to impose some compatibility condition on the initial data.

It should be noted that although the large initial velocity is allowed in ^[12], it excludes large oscillations of the initial density. A natural question arises: where can we establish the global strong solutions to the 3D Cauchy problem of (1.1)–(1.4) not only with large initial velocity but also allowing large oscillations of the initial density? In fact, this is the main aim of this paper.

Before stating the main results, we set

For $1\leqslant r\leqslant \infty$ and $\beta > 0$, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows:

The main result can be stated as follows:

Theorem 1.1. For constant $\bar{\rho}$ and any given number $q\in (3, 6)$, assume that the initial data $(\rho_0, {\mathbf u}_0, {\mathbf w}_0)$ satisfies

Then there exists a positive constant $\varepsilon$, depending only on $\zeta, \mu^{\prime}, \lambda^{\prime}, \bar{\rho}, q, \alpha$, $\beta \triangleq \sup_{[0, \bar{\rho}]} \mu(\rho)$, $\|\nabla \mu(\rho_0)\|_{L^q}$, $\|\nabla {\mathbf u}_0\|_{L^2}$ and $\|\nabla {\mathbf w}_0\|_{L^2}$ such that if

then the problem (1.1)–(1.4) possesses a unique global strong solution $(\rho, {\mathbf u}, {\mathbf w})$ in $\mathbb{R}^3\times (0, \infty)$ satisfying that for any $0 < t < T < \infty$ and $s\in (3, q)$,

Moreover,

and for any $t\geqslant1$,

where $\sigma = 3\sigma_1 (\tfrac{\pi}{2})^{4/3}\|\rho_0\|_{L^{3/2}}^{-1}$ with $\sigma_1 = \min\{\alpha, \mu^{\prime}\}$.

Remark 1.1. It is worth noting that Theorem 1.1 holds for arbitrarily large initial velocity with a smallness only on the initial mass, which generalizes the result of ^[14], where they need the smallness assumption on $\|{\mathbf u}_0\|_{\dot{B}^{3/p}_{p, 1}}$ with $1 < p < 6$.

The rest of this paper is organized as follows: In Section 2, we collect some elementary facts and inequalities that will be used later. Section 3 is devoted to a priori estimates, and Theorem 1.1 is also proved in Section 3.

2.   Preliminaries

In this section, we list some auxiliary lemmas that will be used later. First of all, we start with the local existence of strong solutions that can be obtained from similar arguments as used in ^[13,20], and we omit the details.

Lemma 2.1. Assume that $(\rho_0, {\mathbf u}_0, {\mathbf w}_0)$ satisfies (1.5). Then, there exists a small positive time $T_0$ such that the problem (1.1)–(1.4) has a unique strong solution $(\rho, {\mathbf u}, {\mathbf w})$ on $\mathbb{R}^3\times (0, T_0]$.

Next, the following well-known Gagliardo-Nirenberg inequality will be used more frequently later (see ^[11]).

Lemma 2.2. Let $p\in[2, \tfrac{3s}{3-s}]$ for $s\in[2, 3)$, or $p\in[2, \infty]$ for $s = 3$, and let $q\in(1, \infty), r\in(3, \infty)$. There exists some generic constant $C > 0$ may depend on $s$ and $r$ such that for $f\in L^2\cap D^{1, s}_0$ and $g\in L^q \cap D^{1, r}_0$, we have

and

The following regularity results on the Stokes equations will be used for the derivations of higher-order a priori estimates (see ^[10]).

Lemma 2.3. For constants $q\in(3, 6), \alpha > 0$ and $\beta > 0$, in addition to (1.2), assume that $\mu(\rho)$ satisfies

Then, if ${\mathbf G}\in L^r$ with $r\in (2, q)$, there exists some positive constant $C$ depending only on $\alpha, \beta, r$ and $q$ such that the unique weak solution $({\mathbf u}, P)\in D^{1}_{0, \sigma}\times L^2$ to the following problem

satisfies

and

3.   A priori estimates

In this section, we will establish some necessary a priori bounds of local strong solutions $(\rho, {\mathbf u}, {\mathbf w})$ to the Cauchy problem (1.1)–(1.4), whose existence is guaranteed by Lemma 2.1. Thus, let $T > 0$ be a fixed time and $(\rho, {\mathbf u}, {\mathbf w})$ be the smooth solution to (1.1)–(1.4) on $\mathbb{R}^3\times (0, T]$ with smooth initial data $(\rho_0, {\mathbf u}_0, {\mathbf w}_0)$ satisfying (1.5).

We have the following key a priori estimates on $(\rho, {\mathbf u}, {\mathbf w})$.

Proposition 3.1. There exists some positive constant $\varepsilon_0$ depending only on $q, \zeta, \bar{\rho}, \alpha, \beta, \mu^{\prime}$, $\|\rho_0\|_{L^{3/2}}$, $\|\nabla {\mathbf u}_0\|_{L^2}, \|\nabla {\mathbf w}_0\|_{L^{2}}$, and $\|\nabla \mu(\rho_0)\|_{L^q}$ such that if $(\rho, {\mathbf u}, {\mathbf w})$ is a smooth solution of (1.1)–(1.4) on $\mathbb{R}^3\times (0, T]$ satisfying

the following estimates hold

provided that

The proof of Proposition 3.1 consists of Lemmas 3.1–3.4 and is to be completed by the end of this section. Throughout this section, for simplicity, we denote by $C$ or $C_i (i = 1, 2, \cdots)$ the generic positive constants, which may depend on $q, \zeta, \bar{\rho}, \alpha, \beta, \mu^{\prime}, \lambda^{\prime}, \|\rho_0\|_{L^{3/2}}, \|\nabla {\mathbf u}_0\|_{L^2}$ and $\| \nabla {\mathbf w}_0\|_{L^{2} }$, but are independent of time $T > 0$ and $m_0$.

We begin with the following estimates:

Lemma 3.1. Let $(\rho, {\mathbf u}, {\mathbf w})$ be a smooth solution of (1.1)–(1.4) on $\mathbb{R}^3 \times (0, T]$ satisfying (1.5). Then one has

and

where $\sigma = 3\sigma_1 (\tfrac{\pi}{2})^{4/3}\|\rho_0\|_{L^{3/2}}^{-1}$ with $\sigma_1 = \min\{\alpha, \mu^{\prime}\}$.

Proof. First, Eq (3.4) can be shown by standard arguments (^[8]).

Next, in order to prove (3.5), we multiply (1.1)$_2$ and (1.1)$_3$ by ${\mathbf u}$ and ${\mathbf w}$, respectively, and in integrating the resulting equations by parts over $\mathbb{R}^3$, we get after adding them together and using (1.1)$_4$ that

Integrating (3.7) over $[0, T]$ gives

which, together with (2.1) and (3.4), yields (3.5).

Finally, we notice from (2.1), (3.4), and (3.8) and Hölder's inequality that

where we have used the following fact:

Combining (3.7) with (3.9), one has

where $\sigma = 3\sigma_1 (\tfrac{\pi}{2})^{4/3}\|\rho_0\|_{L^{3/2}}^{-1}$ with $\sigma_1 = \min\{\alpha, \mu^{\prime}\}$. By using Gronwall's inequality, one has

Multiplying (3.7) by $e^{\sigma t}$ and using (3.10) show that

Integrating the above inequality over $[0, T]$ leads to (3.6). □

Remark 3.1. Evidently, we can infer from (1.2) and (3.4) that

Lemma 3.2. Let the condition of (3.1) be in force, then there exists some positive constant $\varepsilon_1$, depending only on $q, \zeta, \bar{\rho}, \alpha, \beta, \mu^{\prime}, \lambda^{\prime}, \|\nabla \mu(\rho_0)\|_{L^q}, \|\nabla {\mathbf u}_0\|_{L^2}$ and $\| \nabla {\mathbf w}_0\|_{L^{2} }$ such that if

then

provided

Moreover, for $i = 1, 2, 3$ and $\sigma$, as in Lemma 3.1, one has

and

Proof. The Eq (1.1)$_1$ can be written as

Multiplying (1.1)$_2$ by ${\bf{u}}_t$ and integrating the resulting equation over $\mathbb{R}^3$, one can deduce from (3.17) that

where we have used the fact that $\Delta {\bf{u}}+{\rm rot}({\rm rot} {\bf{u}}) = \nabla {\rm div} {\bf{u}} = 0$.

Multiplying (1.1)$_3$ by ${\bf{w}}_t$ and integrating by parts over $\mathbb{R}^3$ show that

Combining (3.18) with (3.19), yields

Now, we estimate $I_i(i = 1, 2, 3.)$ as follows: Hölder's inequality, together with (2.1), (2.2), and (3.4), gives

Similarly,

and

Putting (3.21)–(3.23) into (3.20), one has

We know from (1.1)$_2$ that $({\bf{u}}, P)$ satisfies the following system:

Taking ${\mathbf G} = -\rho {\bf{u}}_t-\rho {\bf{u}} \cdot \nabla {\bf{u}}+2\zeta {\rm rot} {\bf{w}}$ and $r = 2$ in (2.3), we deduce from (3.1), (3.4), and Cauchy-Schwarz's inequality that

which gives

Taking (3.25) into (3.24), one deduces from (3.13) that

Integrating (3.26) over $[0, T]$, then using (3.5) and (3.12), shows that

where the positive constant $C_1$ depends only on $q, \zeta, \bar{\rho}, \alpha, \beta, \mu^{\prime}, \lambda^{\prime}, \|\nabla \mu(\rho_0)\|_{L^{q}}, \|\nabla {\mathbf u}_0\|_{L^2}$, and $\| \nabla {\mathbf w}_0\|_{L^{2} }$, but is independent of time $T > 0$ and $m_0$. Taking

Thus, we obtain (3.14).

Multiplying (3.26) by $t$ gives that

Integrating (3.28) over $[0, T]$, we obtain from (3.5) and (3.6) that

which implies that (3.15) holds for $i = 1$. For $i = 2, 3$, we can take a similar approach to obtain the results of (3.15).

Next, multiplying (3.26) by $e^{\sigma t}$, one has

Integrating (3.29) over $[0, T]$, one can deduce from (3.6) that

Thus, we obtain (3.16). □

Lemma 3.3. Under the condition of (3.1), then for $i = 1, 2, 3$,

Moreover, for $\sigma$ as in Lemma 3.1 and $\delta (t) \triangleq \min\{1, t\}$,

Proof. Operating $\partial_t$ to (1.1)$_2$ and (1.1)$_3$, respectively, we infer from (1.1)$_1$ that

and

Multiplying (3.32) and (3.33) by ${\bf{u}}_t$ and ${\bf{w}}_t$, respectively, and then integrating by parts on $\mathbb{R}^3$, we deduce from (3.17) that

Now, we estimate $N_i (i = 1, 2, 3, 4)$ as follows: Thanks to ${\rm div} {\bf{u}} = 0$, we deduce from (2.1), (2.2), (3.4), (3.14) and Cauchy-Schwarz's inequality that

and

Cauchy-Schwarz's inequality gives

The inequalities (2.2) and (3.1), together with Cauchy-Schwarz's inequality, show that

Putting (3.35)–(3.38) into (3.34) and using (3.25) yields

To deal with the last term on the right side of (3.39), we first multiply (1.1)$_3$ by ${\bf{w}}$ and then integrate the resulting equation by parts on $\mathbb{R}^3$ to get that

which implies

On the other hand, one can deduce from (1.1)$_3$ that

which, together with (3.40), (2.1), and (3.4), yields

thus

Taking (3.41) into (3.39), we infer from (3.14) that

Multiplying (3.42) by $t$, one has

which, together with (3.14), (3.15) and Gronwall's inequality, gives that (3.30) holds for $i = 1$. Using the same methods, we can show that (3.30) for $i = 2, 3$.

Next, in order to prove (3.31), we first multiply (3.42) by $e^{\sigma t}$, then

which, combining (3.6), (3.14), and (3.16) with Gronwall's inequality, gives that (3.31) holds. □

Lemma 3.4. Under the condition of (3.1), then

where $3 < r < q$ with $q\in (3, 6)$.

Proof. It follows from (1.1)$_2$, (2.1), (2.2), (2.4), and (3.1) that for any $r\in (3, \min \{6, q\})$

thus

The inequality (3.6), together with Hölder's inequality, shows that

For any $T \in (0, 1]$, we can show from (3.30) and Hölder's inequality that

For $T > 1$, one can deduce from (3.30), (3.31), and Hölder's inequality that

Similarly,

It follows from (3.5), (3.13), and (3.25) that

Due to $3 < r < q, \; q\in (3, 6)$, one has

Integrating (3.45) over $t\in [0, T]$ and using (3.46)–(3.50), we can obtain (3.43). □

With Lemmas 3.1–3.4 at hand, we are in a position to prove Proposition 3.1.

Proof of Proposition 3.1. It follows from (3.17) that

Multiplying the above equation by $q|\nabla \mu(\rho)|^{q-2}\nabla \mu(\rho)$ and integrating the resulting equations on $\mathbb{R}^3$, we can obtain

Due to ${\rm div} {\bf{u}} = 0$, then

Thus

which implies that

which, together with (3.43) and Gronwall's inequality, shows that

provided

Thus, we complete the proof of Proposition 3.1. □

Lemma 3.5. Under the condition of (3.1), then

Proof. Similar to the method of (3.51), we can deduce from (1.1)$_1$ that

Hölder's inequality, together with (1.1)$_1$, (3.14), and (3.53), yields

Thus, we complete the proof of Lemma 3.5. □

Lemma 3.6. Under the condition of (3.1), then for $3 < r < q$ with $q\in (3, 6)$, the following estimates hold:

and

for $\sigma$ as in Lemma 3.1 and $\delta(t)$ as in Lemma 3.3.

Proof. It follows from (3.14), (3.15), (3.25), (3.30), and (3.41) that

thus

Hence, we can use the same methods to obtain (3.55).

By virtue of $r\in (3, q)$ with $q\in (3, 6)$, one can obtain from (3.44) that

which, together with (3.5), (3.15), (3.25), (3.30), (3.41), and (3.56), yields

Thus, we complete the proof of Lemma 3.6. □

Proof of Theorem 1.1. Similar to the standard arguments in ^[10,18,20], with all the a priori estimates established in Sections 3 at hand, we can immediately obtain our main results. □

Use of AI tools declaration

The author declares she has 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 that she has no conflict of interest.