Asymptotic behavior of the one-dimensional compressible micropolar fluid model

1.   Introduction and the main theorem

In this paper, we consider the one-dimensional compressible micropolar fluid model in Lagrange coordinates:

where $t>0$ is time, $x\in \Omega\subseteq \mathbb{R}$ denotes the Lagrange mass coordinate, $v = v(x, t)>0$, $u = u(x, t)$, $w = w(x, t)$, $\theta = \theta(x, t)>0$, $e$ and $P$, which represent the specific volume, fluid velocity, micro-rotation velocity, absolute temperature, internal energy and pressure, respectively. Moreover, $\mu>0$ is the viscosity coefficient, $\kappa>0$ is the heat conductivity coefficient, $R>0$ is the specific gas constant, $\nu>0$ and $\lambda>0$ are the coefficients of micro-viscosity, $c_v$ is the heat capacity at constant volume. The model of micropolar fluid was first introduced by Eringen [7] in 1966. This model can be used to describe the motions of a large variety of complex fluids consisting of dipole elements such as the suspensions, animal blood, liquid crystal, etc. For more physical background on this model, we can refer to [8,15].

When $w = 0$, the system (1.1) reduces to the standard compressible Navier-Stokes equations. It has attracted great interest among many analysts and there have been many important developments. Kazhikhov and Shelukhin [12] firstly obtained the global existence of solutions in bounded domains for large initial data. Then, Jiang [10,11] studied the large-time behavior of solutions to the compressible Navier-Stokes system in unbounded domains for large initial data. He proved the specific vomule was pointwise bounded from below and above independent of both space and time. Recently, Li and Liang [13] proved the large-time behavior of solutions to the initial and initial boundary value problems with large initial data. They showed that the temperature was bounded from below and above independent of both space and time. See also [5] for the spherically symmetric and cylindrically symmetric non-barotropic flows.

When $w\neq0$, we get the micropolar fluid model. Now more and more mathematicians devoted to the research of micropolar fluid model (1.1). For the incompressible fluid, we refer to [3] and references therein. For the complete fluid, Mujakovic [16,17] obtained a series of results about the local, global in time existence theorem and regularity of solutions for the model with homogeneous boundary condition in one dimension. Next, Chen [2] proved the global existence of strong solutions for the initial boundary problem in one dimension that vacuum can be allowed initially. Yin [19] established the stationary solutions to the inflow problem in a half-line, the time asymptotically stability for the $H^{1}$ solutions was also obtained. Recently, Cui and Yin [6] obtained the convergence rates of global solutions toward corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. Qin et al. [18] studied the global existence and asymptotic behavior of $H^1$ solutions to the Cauchy problem of one-dimensional model with the weighted small initial data. Based on the assumption $\kappa(\theta) = O(1) (1+\theta^q)$ with $q\geq0$, Feng and Zhu [9] considered the existence of global classical solution with large initial data and vacuum. For three-dimensional micropolar fluid model, Chen et al. [4] gave the global weak solutions with discontinuous initial data and vacuum. Liu and Zhang [14] obtained the optimal time decay of the 3-dimensional compressible micropolar fluid model.

In this paper, motivated by the result of [13], we get asymptotic behavior of the one-dimensional compressible micropolar fluid model with large initial data. We supplement the system (1.1) with the initial condition

and three types of far-field and boundary conditions:

(1) Cauchy problem

(2) Boundary and far-field conditions for $\Omega = (0, \infty)$

(3) Boundary and far-field conditions for $\Omega = (0, \infty)$

We now state the main result of this paper as follows:

Theorem 1.1. Consider the initial value problem given by (1.1)-(1.3), and initial-boundary value problem given by (1.1), (1.2), (1.4) and (1.1), (1.2), (1.5). Assume that $v_0(x)>0$, $\theta_0(x)>0$, $v_0-1$, $u_0$, $w_0$, $\theta_0-1$ $\in H^{1}(\Omega)$ and the initial data are compatible with boundary conditions. Then there exists a unique global strong solution $( v , u, w, \theta)\in C([0, \infty), H^{1}(\Omega))$, $(v-1, u, w, \theta-1)\in L^\infty([0, \infty), H^{1}(\Omega))$, and there exists positive constants $C_{1}$ and $C$ depending only on $\mu$, $\kappa$, $R$, $c_{v}$, $\nu$ satisfying

and for any $t\geq0$, $p>2$

Remark 1. Compared with the work [13], we have to deal with the strong couple of the micro-rotation velocity $w$ and the fluid motion, which will lead to some difficulties. In particular, we solve it by obtaining some new estimates (2.16), (2.31). We also obtain the asymptotic behavior of $w(x, t)$.

2.   Proof of Theorem 1.1

2.1.   Preliminaries

Lemma 2.1. (Energy inequality) Under the conditions of Theorem 1.1, the following inequality holds

and there are $a_i(t), b_i(t)\in [i, i+1]$ satisfying

where $\alpha_1, \alpha_2$ are two positive roots of the equation $y-\ln y -1 = \frac{C_0}{\min\{R, c_v\}}$, $i\in \mathbb{N}$ for the initial boundary value problems and $i\in \mathbb{Z}$ for the Cauchy problem.

Proof. The above estimates can been obtained by the similar arguments as that in [10].

Lemma 2.2. Under the conditions of Theorem 1.1, there holds

and

Proof. Multiplying $(1.1)_3$ by $w$, using the boundary condition of $w$, one directly gets (2.4).

Multiplying $(1.1)_3$ by $w^3$, using the boundary condition of $w$, one gets (2.5).

2.2.   Uniform estimates of $v$

In this subsection, the specific volume $v(x, t)$ is proved to be bounded from below and above, independent of both time and space.

Lemma 2.3. Under the hypotheses of Theorem 1.1, it holds that

For more detailed proof of Lemma 2.3, refer to [5,10,11] and references therein.

2.3.   Uniform estimates of $\theta$

Lemma 2.4. Under the conditions of Theorem 1.1, let $(v, u, w, \theta)$ be a solution to the system (1.1) on ${\Omega}\times[0, T]$, it holds for any $T>0$

Proof. Motivated by [13], the proof will be divided into three steps.

Step 1. At first, for any $t\geq0$ and $a>1$, denoting

and we derive from (2.1) that

Using $(1.1)_1$, $(1.1)_3$, $(1.1)_5$, we rewrite $(1.1)_4$ as

Multiplying (2.8) by $(\theta-2)_+ = \sup\{\theta-2, 0\}$ and integrating the result over $\Omega\times[0, T]$, we have

Next, multiplying $(1.1)_2$ by $2 u(\theta-2)_+$ and integrating the result over $\Omega\times[0, T]$, we obtain

Then multiplying $(1.1)_3$ by $2 w(\theta-2)_+$ and integrating the result over $\Omega\times[0, T]$, yields

Adding the above three integral expressions together and using (2.8) to get

Now the main task is to estimate $I_{1}, I_{2}, I_{3}......, I_{12}$ term by term.

For $I_{1}$, using Cauchy inequality and (2.6), one obtains

It follows from Cauchy inequality and (2.6) that for any $\epsilon>0$

where we have used the fact that

Then for $I_3$, it follows from Cauchy inequality and (2.12)

For $I_4$,

For any $\eta>0$, set

For $I_5$, we immediately get for any $\epsilon>0$

It follows from Cauchy inequality that

For $I_8$,

For $I_9$, using (2.5), we have

For the terms $I_{10}$-$I_{12}$, it follows from Cauchy inequality that

and

Noticing that

Then

Step 2. Multiplying $(1.1)_3$ by $2u^2 w$ and integrating the result over $\Omega\times [0, T]$, we have

Using $(1.1)_2$, we obtain

What's more, it follows from (2.1) and (2.6) that for any $\alpha\in[2, 3]$,

Then we can estimate $J_{j}$($j = 1, 2, 3, 4$),

and

Therefore, we have

And similarly,

Adding all the above inequalities, one obtains

Step 3. We estimate the last term on the right hand side of (2.17).

Then

The proof of Lemma 3.4 is complete.

2.4.   Uniform estimates on the derivatives of $v$, $u$, $w$, $\theta$

Lemma 2.5. There exists a positive constant $C$ such that for any $T>0$, it holds that

Moreover,

Proof. The process will be divided into five steps:

Step 1. Integrating $(1.1)_2\times \frac {v_{x}}{v}$ over $\Omega$, we get by using $(1.1)_1$

Integrating the above equality on $[0, T]$, we have

It follows from Cauchy inequality, (1.6), (2.1), (2.6), (2.7) and (2.18) that

Step 2. Integrating $(1.1)_2\times(-u_{xx})$ over $\Omega$ leads to

Using (2.7), (2.22) and Sobolev inequality, we get

then by using (2.23), one has

Step 3. Integrating $(2.8)\times(-\theta_{xx})$ over $\Omega$ leads to

then using Cauchy inequality, (2.7), (2.22), (2.25), Sobolev inequality, we get

Hence we obtain by using (2.26) that

Step 4. Integrating $(1.1)_3 \times (-w_{xx})$ over $\Omega$, we have

using (2.1), (2.22) and Lemma 2.2, one has

Then, by using (2.29), we get

Step 5. Using Sobolev inequality and (2.7), for any $0\leq t\leq T$

Combining (2.28), yields

It implies that there exists a positive constant $C$ for any $(x, t)\in\bar{\Omega}\times[0, T]$

Hence, putting together (2.22), (2.25), (2.28), (2.31) and (2.33), we finish the proof of Lemma 2.5.

2.5.   Large-time behavior of global solutions

Lemma 2.6. Under the conditions of Theorem 1.1, it holds that

for any $p>2$, there also exists a positive constant $C_2$

Proof. First, it follows from (2.7), (2.20), (2.23), (2.24), (2.26), (2.27), (2.29) and (2.30) that

which directly gives

By applying (2.36) to (2.32), we get

Hence, there exists some $T^{\ast}>0$

Combining (2.20), leads to

Then combining $(1.1)_1$ and (2.20), one has

which together with (2.38) implies

Therefore, combining (2.1), (2.7), (2.36), (2.39) and (2.15), we can get (2.34).

Finally, we will establish the lower bound of $\theta$. According to [1,12], there exists a constant $C_{3}>2$

which together with (2.37), yield

Combining (2.33) and (2.35), we choose $C\triangleq\max\{{C_{2}, C_{3}e^{C_{3}T^{\ast}}}\}$. The proof of Lemma 3.6 is finished.