Global solvability of 3D non-isothermal incompressible nematic liquid crystal flows

1.   Introduction

The motion of incompressible non-isothermal nematic liquid crystal flows can be governed by the following simplified version of the Ericksen-Leslie equations (see ^[12,13])

which is equipped with the following initial conditions

Here, $\rho$, $u$, $\theta$ and $d$, $p$ are the density, velocity, absolute temperature, macroscopic average of the nematic liquid crystal orientation field and pressure respectively. The $\nabla d\odot\nabla d$ denotes a matrix whose $ij$-th entry $(1\le i, j\le N)$ is $\partial_{x_i}d\cdot\partial_{x_j}d$. Indeed, $\nabla d\odot\nabla d = (\nabla d)^\top\nabla d$, where $(\nabla d)^\top$ denotes the transpose of the $3\times 3$ matrix $\nabla d$. $\mathfrak{D}(u)$ denotes the deformation tensor given by

The constant $\mu > 0$ is the viscosity coefficient. The positive constants $c_v$ and $\kappa$ are respectively the heat capacity and ratio of the heat conductivity coefficient to the heat capacity.

The full Ericksen-Leslie model and the simplified version are macroscopic continuum descriptions of the time evolution of the materials, under the influence of both the flow velocity field $u$ and the microscopic orientation configurations $d$ of rod-like liquid crystals. Mathematically, system (1.1) is a strongly coupled system between the nonhomogeneous incompressible Navier-Stokes equations of fluid dynamics and the transported heat flows of harmonic maps. When $d$ is a constant vector and $|d| = 1$, (1.1) reduces to the nonhomogeneous heat-conducting Navier-Stokes equations, which have been discussed in numerous studies on the global well-posedness of strong solutions(please see ^{[1,3,14,15,16,18,19,20]} and references therein). Due to the strong coupling terms and strong nonlinear terms, the system (1.1) the well-posedness is harder to study than Navier-Stokes equations.

If we don't take into account $(1.1)_3$, the system (1.1) becomes the classical inhomogeneous incompressible nematic liquid crystal flows. In the presence of a vacuum, under the following compatibility condition

for some $(p_0, g)\in H^1\times L^2$, Wen-Ding ^[17] first established the local existence and uniqueness of a strong solution. Later, Li ^[4] and Ding et al. ^[2] respectively investigated the global existence of a strong solution to the 3D initial boundary value problem and to the 3D Cauchy problem with reasonably small data; they extended Wen-Ding's result^[17] to the global one. At the same time, Li ^[4] considered the case in 2D-bounded domains (see also ^[5]). It should be pointed out that the crucial techniques of proofs in ^[2,4,5] cannot be directly adapted to the situation for the Cauchy problem of the 2D equations. One reason is that when $\Omega = \Bbb R^2$ becomes unbounded, the standard Sobolev embedding inequality is critical, and it seems difficult to bound the $L^r$-norm ($r > 2$) of the velocity $u$ just in terms of $\|\sqrt{\rho}u\|_{L^2(\Bbb R^2)}$ and $\|\nabla u\|_{L^2(\Bbb R^2)}$. Motivated by ^[10], by introducing a weighted function to the density, as well as the Hardy-type inequality proposed in ^[7] by Lions, the $\|\rho^\alpha u\|_{L^r}$ $(r > 2, \alpha > 0)$ could be controlled in terms of $\|\sqrt{\rho}u\|_{L^2(\Bbb R^2)}$ and $\|\nabla u\|_{L^2(\Bbb R^2)}$.Thus, Liu et al. ^[6,8] generalized the results of ^[4,5] to the whole space $\Bbb R^2$.

Very recently, the author of ^[9] applied global well-posedness to the 2D Cauchy problem with a vacuum and large initial data. However, it is rather difficult to investigate the global well-posedness and dynamical behaviors of non-isothermal incompressible nematic liquid crystal flows with large initial data in $\Bbb R^3$ using the same method. Since the Sobolev embedding inequality is critical, we cannot derive the a priori estimates for the strong solutions with large initial data, which is the key to extending the global strong solution by Lemma 2.1. Fortunately, motivated by ^[4], the global strong solutions of the systems (1.1) and (1.2) can be established using some small norm of the initial. But the priori estimates of ^[4] depend on the boundness of the domain due to the Poincaré inequality, which fails to deal with the situation here. On the other hand, there are also other interesting studies on the large time behavior of solutions, which is absent in ^[2,4]. Thanks to ^[3], which established the global existence and large time behavior of strong solutions to the 3D Cauchy problem of Navier-Stokes equations using exponential time-weighted energy estimates. However, due to the strong coupling terms and strong nonlinear terms $\nabla d\odot \nabla d$, $u\cdot\nabla d$ and $|\nabla d|^2d$ in the system (1.1), we cannot directly adapt their method to our case. Thus, the main purpose of the present study was to investigate the global existence and large time behavior of strong solutions with large oscillations and a vacuum to the problems $(1.1)$ and $(1.2)$, provided that the scaling invariant quantity is properly small which generalized the corresponding results ^[3,4] to the Cauchy problem of non-isothermal incompressible nematic liquid crystal flows.

Before formulating our main results, we will first explain the notations and conventions used throughout this paper. For simplicity, we set

For $1\le r\le \infty$, $k\ge 1$, the Sobolev spaces are defined in a standard way.

Our main results in this paper are listed in the following Theorem 1.1.

Theorem 1.1. Assume that the initial data $(\rho_0, u_0, \theta_0, d_0)$ satisfies

and the compatibility condition

with $p_0\in H^1$ and $g_1, g_2\in L^2$.Let $\Bbb K_0$ be the following constant

Then, there is a positive constant $\epsilon_0$, which depends only on $\bar{\rho}$ and initial data, such that if

then the problems $(1.1)$ and $(1.2)$ have a unique global solution $(\rho, u, \theta, d)$ on $\Bbb R^3\times(0, \infty)$ satisfying

Moreover, there exists some positive constant $C$ depending only on $\bar{\rho}$ and initial data such thatfor all $t\ge 1$,

Remark 1.1. When temperature field $\theta\equiv0$, System $(1.1)$ reduces to the classical incompressible nematic liquid crystal flows. Compared to ^[4] for the 3D-bounded domain, some new difficulties occur. First, Poincare's inequality fails for 3D Cauchy problem, which is key to estimating $\|u\|_{L^2}$; one way to overcome this is to estimate $\|\sqrt{\rho}u\|_{L^2}$ and $\|\nabla u\|_{L^2}$. Furthermore, the higher-order estimates of the smooth solution is independent on $T$, which is different from ^[4]. Since the corresponding estimates in ^[4] depend on $T$, we have improved the priori estimates of ^[4]. In particular, we also extend the priori estimates of ^[4] to the non-isothermal case. To the best of our knowledge, Theorem 1.1 is the first result on global well-posedness for a model of multi-dimensional non-isothermal incompressible nematic liquid crystal flows in a vacuum.

Remark 1.2. Using the time-weighted a priori estimates, we extend Zhong's corresponding result ^[19] to incompressible nematic liquid crystal flows by studying the global well-posedness of the Cauchy problem of inhomogeneous heat-conducting Navier-Stokes equations. Moreover, we derive the algebraic decay estimates of the solution, which is lacking in ^[19].

Remark 1.3. As mentioned above, He et al. ^[3] obtained the exponential decay-in-time property for the 3D Cauchy problem of inhomogeneous Navier-Stokes equations with a vacuum [that is, $(1.1)$ with $d = 1$ and $\theta = 0$]; however, it seems impossible to derive such a decay property due to the sructural characteristics of $(1.1)$, especially given that $\rho_0\in L^\frac32(\Bbb R^3)$ is required. Thus, the the macroscopic average of the nematic liquid crystal orientation field acts as some significant roles on the large time behaviors of velocity and the temperature.

Remark 1.4. It should be noted that $(1.6)$ is independent of the temperature. The results indicate that the global existence for inhomogeneous incompressible heat conducting models of viscous media is similar to the nonhomogeneous incompressible nematic liquid crystal flows, and therefore does not depend on further sophistication of the heat conducting model.

The framework of the proof of Theorem 1.1 is the continuation argument (see ^[4]). We should point out that the crucial techniques used in ^[4] cannot be adapted to the situation treated here, since their arguments depend heavily on the boundedness of the domains. Moreover, compared to ^[3,19], the proof of Theorem 1.1 is much more involved due to the strong coupling terms and strong nonlinear terms $\nabla d\odot \nabla d$, $u\cdot\nabla d$ and $|\nabla d|^2d$. To overcome the difficulties stated above, we need some new ideas. First, we attempted to get time-in-uniform (weighted) estimates on the $L^\infty(0, T; L^2)$-norm of the gradient of velocity, absolute temperature and the gradient of the macroscopic average of the nematic liquid crystal orientation field. We found that the key way to control the terms $\int|u|^2|\nabla^2d|^2dx$, $\int|\nabla d|^2|\nabla u|^2dx$ and $\int|\nabla d|^2|\nabla^2d|^2$. Thus, we need to achieve higher integrability of the macroscopic average of the nematic liquid crystal orientation field. Motivated by ^[3,19], we assume the priori hypothesis (3.1). Hence, the key step is to complete the proof of the a priori hypothesis, that is, to show (3.2). Based on the regularity properties of the Stokes system and elliptic equations, we can obtain the desired bounds, provided that (1.7) is suitably small. Next, the crucial dissipation estimate of the form $\int_0^T\|\nabla\theta\|_{L^2}^2dt$ (see Lemma 3.5) is important to derive the time-weighted estimate of $\|\sqrt{\rho}\theta_t\|_{L^2}^2$. However, this argument fails for the situation here, since their estimates dependent on $T$. To overcome this difficulty, we multiply the momentum equation$(1.1)_2$ by $u\theta$ to recover good bounds for the temperature $\theta$ [see (3.81)]. Once these key estimates are obtained, we can obtain the desired global high-order a priori estimates of the solution.

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 the a priori estimates. Finally, we will give the proof of Theorem 1.1 in Section 4.

2.   Preliminaries

We begin with the local existence and uniqueness of strong solutions whose proof can be performed by using standard energy methods (see ^[1,17]).

Lemma 2.1. Assume that $(\rho_0, u_0, \theta_0, d_0)$ satisfies $(1.4)$ and $(1.5)$. Then there existsa small time $T_0 > 0$ and a unique strong solution $(\rho, u, \theta, d, p)$ to the problems $(1.1)$ and $(1.2)$ in $\Bbb R^3\times (0, T)$ satisfying $(1.8)$.

Next, we have some regularity results for the Stokes equations, which have been proven in ^[3].

Lemma 2.2. For $r\in (1, \infty)$, if $F\in L^\frac65\cap L^r$, there exists somepositive constant C depending only on r such that the unique weaksolution $(u, p)\in D^1\times L^2$to the following Stokes system

satisfies

and

Next, the following Gagliardo-Nirenberg inequalities will be stated(see ^[11] for the detailed proof).

Lemma 2.3. Assume that $f\in H^1$ and $g\in L^q\cap D^{1, r}$ with $q > 1$ and $r > 3$. Then for any $p\in [2,6]$, thereexists a positive constant $C$, depending only on $p$, $q$ and $r$, such that

3.   A priori estimates

In this section, we will establish some necessary a priori estimates, which together with the local existence (cf. Lemma 2.1) will complete the proof of Theorem 1.1. To this end, we let $(\rho, u, d, \theta, p)$ be a strong solutions of (1.1) and (1.2) in $\Bbb R^3\times[0, T]$. For simplicity, we use the letters $C$, $c_i$ and $C_i(i = 1, 2, \ldots)$ to denote some positive constant that is dependent on $\bar{\rho}$ and initial data, but are independent of $T$. We also sometimes write $C(\alpha)$ to emphasize the dependence on $\alpha$.

We first aim to get the following key a priori estimates on $(\rho, u, d, \theta, p)$.

Proposition 3.1. Assume that

there exists some small positive constant $\epsilon_0$ depending only on $\bar{\rho}$ and initial data such that if $(\rho, u, d, \theta, p)$ is a smooth solution of $(1.1)$ and $(1.2)$ on $\Bbb R^3\times (0, T]$ satisfying

then the following estimate holds

Moreover, we have

Proof of Proposition 3.1. (1) Multiplying the transport equation $(1.1)_1$ by $q\rho^{q-1}$ for $\forall q > 1$, integrating by parts and using $(1.1)_5$, we have

It derives

(2) Multiplying $(1.1)_2$ by $u$, and integrating by parts over $\Bbb R^3$, we have

Using the fact that $|d| = 1$ and integrating by parts, we infer from $(1.1)_4$ and Hölder's and Gagliardo-Nirenberg inequalities that

which together with (3.1) and (3.5) yield

This gives rise to

provided

Then, integrating (3.8) with respect to $t$, we arrive at

(3) Multiplying $(1.1)_2$ by $u_t$, and integrating by parts over $\Bbb R^3$, it follows that

(4) Multiplying $(1.1)_4$ by $-\nabla\Delta d$ and then integrating by parts over $\Bbb R^3$, it follows from Hölder's and Gagliardo-Nirenberg inequalities that

It follows from Hölder's, Young's and Gagliardo-Nirenberg inequalities that

(5) Notice that $(\rho, u, d)$ satisfies the following Stokes system

Applying the standard $L^p$-estimate to the above system ensures that for any $p\in(1, \infty)$,

from which, after using (3.1), (3.17), and Gagliardo-Nirenberg inequality, we have

that is

Applying the gradient operator to $(1.1)_4$, we get

It follows from (3.20) that

By assumption, it follows from Hölder's and Young's inequalities that

(6) Multiplying (3.20) by $4|\nabla d|^2\nabla d$ and integrating by parts yields

and hence

Thus, by choosing some constant $c_1$ suitably large such that

then applying $(3.10)+(3.11)+(3.21)+(3.23)\times c_1$ and using (3.22), (3.19) and (3.24), and Young's inequality, we then obtain

By using the Gronwall inequality, (3.9), and choosing $\delta$, $\varepsilon$ suitably small, and noticing that $\|\sqrt{\rho_0}u_0\|_{L^2}^2 +\|\nabla d_0\|_{L^2}^2\le \sup\limits_{0\le s\le t}(\|\sqrt{\rho}u\|_{L^2}^2+\|\nabla d\|_{L^2}^2)$, we have

provided $\Bbb K_0\le \epsilon_1$. Thus, we have

(7) Combining (3.8) and (3.27), we have

provided

As a consequence, we directly obtain (3.2). The proof of Proposition 3.1 is finished.

Lemma 3.1. Under the conditions of Proposition 3.1, it holds that for $i\in\{0, 1\}$

Proof. 1) Using Hölder's and Gagliardo-Nirenberg inequalities, we have

which yields

This along with Gronwall's inequality, (3.9) and (3.27) yields the desired (3.29).

2) It follows from (3.5) that

Multiplying the above inequality by $t^\frac12$ and integrating it over $[0, T]$, we have

Multiplying (3.25) by $t$, we obtain the desired (3.31) after using Gronwall's inequality. Then, we completed the proof of Lemma 3.1.

Lemma 3.2. Under the assumption of Theorem 1.1, it holds that for $i\in\{1, 2\}$

Proof. (1) Differentiating $(1.1)_2$ with respect to the time variable $t$ gives

Multiplying the above equality by $u_t$ and integrating the resulting equality by parts over $\Bbb R^3$, we deduce after using $(1.1)_1$ that

The terms on the right-hand side of (3.38) can be bounded as follows. It follows from Hölder's, Young's and Gagliardo-Nirenberg inequalities that

Substituting $I_{21}-I_{24}$ into (3.38), we have

(2) Differentiating (3.20) with respect to the time variable $t$ gives

Multiplying (3.40) by $\nabla d_t$, and integrating the resulting equality over $\Bbb R^3$, we find that

Applying Hölder's and Gagliardo-Nirenberg inequalities, and (3.27), we have

Inserting the estimates of $I_{4i} (i = 1, \ldots, 5)$ into (3.41), it follows that

Now, the inequality of (3.42) added to (3.39), we infer that

(3) It follows from (3.20) and Hölder's and Gagliardo-Nirenberg inequalities that

that is

which given (3.9), (3.3), (3.29) and (3.55) yields

and

Thus, multiplying (3.43) by $t^i$ and using Gronwall's inequality, by (3.9), (3.27) and (3.45), we immediately arrive at

(4) We infer from (3.20), Hölder's and Gagliardo-Nirenberg inequalities and $|d| = 1$ that

that is

which together with (3.9), (3.29), (3.55) and (3.60) yields that

(5) Differentiating $(1.1)_4$ with respect to $t$, multiplying the resulting equality by $d_t$ and then integrating by parts over $\Bbb R^3$, we arrive at

Applying Hölder's and Gagliardo-Nirenberg inequalities, we derive

Hence, one gets

Multiplying it by $t$ and applying Gronwall's inequality, we have

Finally, combining (3.46), (3.49) and (3.47), we have the desired (3.36). Thus, we finished the proof of Lemma 3.2.

Lemma 3.3. Under the assumption of Theorem 1.1, it holds that

Proof. 1) Given (3.25), one obtains from (3.44) that

Multiplying it by $t^\frac32$, integrating it over $[0, T]$ and then choosing $\delta$ reasonably small, we derive

2) Multiplying (3.39) by $t^\frac52$ and using Gronwall's inequality, we infer from (3.39) that

Combining (3.60), (3.59) and (3.44), we have the desired (3.57).

Lemma 3.4. Under the assumption of Theorem 1.1, it holds that

Proof. (1) It follows from Lemma 2.2 and Gagliardo-Nirenberg and Hölder's inequalities that for $r\in (3, 6)$,

which directly deduces that

On the one hand, it follows from (3.9), (3.3) and (3.56) that for $0 < t\le 1$,

On the other hand, using (3.56), (3.36), (3.29) and (3.55), we obtain that

which leads to

Combining (3.63) and (3.65), one obtains

(2) Differentiating the continuity equation $(1.1)_1$ with respect to $x_i$ gives rise to

Multiplying (3.67) by $s|\rho_{x_i}|^{s-2}\rho_{x_i}\ (s = \{2, 6\})$ and integrating the resulting equation over $\Bbb R^3$ gives

It follows from Gronwall's inequality and (3.66) that

Noticing the following facts

(3) Taking $\nabla$ operator to $(1.1)_4$, we get

Using the $L^2$-estimates of an elliptic system, we derive

which yields to

multiplying $t^2$ by (3.73), and integrating the resultant in ($0, T$), and using (3.3), (3.21), (3.31), (3.34)and (3.36), one obtains

(4) According to (3.17), we have

This together with (3.34), (3.36), (3.36) and (3.3) gives

and

This ends the proof of Lemma 3.4.

Lemma 3.5. Under the assumption of Theorem 1.1, it holds that for $i\in\{0, 1\}$,

Proof. 1) Multiplying $(1.1)_3$ by $\theta$ and integrating by parts, one obtains

In fact, multiplying $(1.1)_2$ by $u\theta$ and integrating the resulting equation over $\Bbb R^3$ gives

where we have used the following fact that

It follows from (3.9), (3.3), Hölder's inequality and the Gagliardo-Nirenberg inequality that

Collecting the above estimates, choosing $\delta$ to be reasonably small and applying $\Bbb K_0\le \epsilon_3: = \min\Big\{\epsilon_2, \Big(\frac{1}{4C_3}\Big)^4\Big\}$, we get that

Integrating the above inequality with respect to $t$, after using (3.9), (3.3) and (3.45) one obtains that

(2) Multiplying (3.83) by $t^\frac{1}{2}$, we derive from (3.3), (3.55), (3.56), (3.29), (3.45), (3.31) and (3.84) that

(3) In view of the standard estimate for an elliptic system, one obtains

which leads to

(4) Multiplying $(1.1)_3$ by $\theta_t$ and integrating by parts, we have

By using (3.4), (3.86) and Hölder's, Young's and the Gagliardo-Nirenberg inequalities, we have

Inserting $K_i(i = 1, 2, 3)$ into (3.87) and choosing $\delta$ suitably small, we get

where

satisfies

Then, the desired (3.79) follows from Gronwall's inequality, (3.29), (3.36), (3.34), (3.85), (3.86), (3.88) and (3.90).

Lemma 3.6. Under the assumption of Theorem 1.1, it holds that

Proof. (1) Applying the operator $\partial_t$ to $(1.1)_3$ and a series of direct computations yields

Multiplying (3.93) by $\theta_t$ in $L^2$ and integrating by parts over $\Bbb R^3$ yields

It follows from (3.4), Hölder's inequality and the Gagliardo-Nirenberg inequality that

Substituting $Z_i(i = 1, 2, \cdots, 5)$ into (3.94) and choosing $\delta$ suitably small, we have

(2) Multiplying the above inequality by $t^\frac{5}{2}$, integrating the result with respect to $t$ and using Gronwall's inequality, we have

(3) It follows from (3.9), (3.55), (3.29), (3.36), (3.31), (3.78) and (3.79) that

and

Additionally,

Thus, we completed the proof of the lemma.

4.   Proof of Thorem 1.1

Based on Lemma 2.1, there exists a $T_0 > 0$ such that the magneto-micropolar systems (1.1) and (1.2) have a unique local strong solution $(\rho, u, \theta, d, p)$ in $\Bbb R^3\times[0, T_0]$. To prove Theorem 1.1, it suffices to show that the local solution can be extended to be a global one. To do this, we assume from now that $\Bbb K_0\le \epsilon_0$ holds.

Set

We claim that

Otherwise, assume that $T^* < \infty$. By virtue of Lemmas 3.1 and 3.6 and Proposition 3.1, it holds that $(\rho, u, \theta, d, p)|_{t = T^*}$ satisfies (1.4) and (1.5). Thus, Lemma 2.1 implies that there exists some $T^{**} > T^*$, such that $(\rho, u, \theta, d, p)$ can be extended to a strong solution of (1.1) and (1.2) in $\Bbb R^3\times [0, T^{**})$, which contradicts (4.1). Hence, (4.2) holds.

5.   Conclusions

In this study, we were concerned with an initial value problem related to non-isothermal incompressible nematic liquid crystal flows in $\Bbb R^3$. Using some time-weighted a priori estimates, we have proven the global existence of a strong solution provided that $\Big(\|\sqrt{\rho_0}u_0\|_{L^2}^2+\|\nabla d_0\|_{L^2}^2\Big)\Big(\|\nabla u_0\|_{L^2}^2+\|\nabla^2d_0\|_{L^2}^2\Big)$ is suitably small. Furthermore, we have also obtained the large time behavior of the solutions.

Acknowledgments

The work was supported by the NSF of China (11901288), Postdoctoral Science Foundation of China (2021M691219), Scientific Research Foundation of Jilin Provincial Education Department (JJKH20210 873KJ and JJKH20210883KJ), Natural Science Foundation of Changchun Normal University and doctoral research start-up fund project of Changchun Normal University.

Conflict of interest

The authors declare that there are no conflicts of interest.