Global solution to the compressible non-isothermal nematic liquid crystal equations with constant heat conductivity and vacuum

This paper considers the initial-boundary value problem of the one-dimensional full compressible nematic liquid crystal flow problem. The initial density is allowed to touch vacuum, and the viscous and heat conductivity coefficients are kept to be positive constants. Global existence of strong solutions is established for any H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{2}$\end{document} initial data in the Lagrangian flow map coordinate, which holds for both vacuum and non-vacuum case. The key difficulty is caused by the lack of the positive lower bound of the density. To overcome such difficulty, it is observed that the ratio of ρ0(y)ρ(t,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{\rho _{0(y)}}{\rho (t,y)}$\end{document} is proportional to the time integral of the upper bound of temperature and vector director field, along the trajectory. Density weighted Sobolev type inequalities are constructed for both temperature and director field in terms of ρ0(y)ρ(t,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{\rho _{0(y)}}{\rho (t,y)}$\end{document} and small dependence on their dissipation estimates. Besides this, to deal with cross terms arising due to liquid crystal flow, higher order priori estimates are established by using effective viscous flux.


Introduction
Liquid crystal material can be known as an intermediate phase between solid and liquid. It flows like liquid and exhibits additional microscopic structural properties. For example, rigid polymers solution, DNA, and many other materials show phase change for different ranges of temperature. On the base of structural properties, liquid crystals are categorized as nematic, sematic, and cholesteric phases. Nematic liquid crystal are made up of rod-like molecules, and their structure induces preferred average directional order. The historic example of nematic phase is N-p-methoxybenzlidene-p-butylaniline (MBBA) which appears in between 20°C and 47°C temperatures. The kinematic behavior of nematic liquid crystals due to the strong coupling between the microscopic and macroscopic interaction field is very complicated and interesting. The hydrodynamic theory for nematic liquid crystal is derived by extending the static theory in generalizations of body and surface forces as in the initial derivation of macroscopic hydrodynamics. The system that describes the hydrodynamics of nematic liquid crystal was proposed during the period between 1958 and 1968 by Ericksen [1,2] and Leslie [3] and reduces to the Oseen-Frank theory [4] in the static case. It is a macroscopic continuum description of materials evolution under the effects of both the flow velocity field u(x, t) and the average molecular orientation d(x, t) (the microscopic configuration) of rod-like liquid molecules and their mutual interactions, whereas d(x, t) is a unit vector. Recently, there has been modeling study of a generalized non-isothermal Ericksen-Leslie system by Hieber and Prüss [5] (incompressible non-isothermal case), [6] (compressible non-isothermal case) with consistency on the laws of thermodynamic, and De Anna-Liu [7] derived the generalized compressible nonisothermal Ericksen-Leslie system with full Oseen-Frank energy density. We also refer to [8] where the author derived more sophisticated thermodynamically consistent models of nematic liquid crystal flows both in tensorial and vectorial forms by using the conservation dissipation formalism.
Suppose that ρ(x, t), u(x, t), d(x, t), and θ (x, t) are mass density, fluid velocity vector field, director vector field (represents preferred average directional order), and temperature, respectively. Analogous to one constant approximation model [9], the couple system of thermally driven fluid flow and director field satisfies the following conservation laws: ∂ t (ρθ) + div(ρuθ ) + p(ρ, θ ) div u + div q = S : ∇u + | d + |∇d| 2 d| 2 , whereas the Cauchy stress tensor S is defined as S = μ ∇u + ∇u + λ (div u)I, the viscosity coefficients μ and λ satisfy μ > 0, 2μ + 3λ ≥ 0, and ∇u is the transpose of the matrix ∇u, I is a 3 × 3 identity matrix. The notation ∇d ∇d is a 3 × 3 matrix with (i, j)th entries, given by ∂ i d · ∂ j d (1 ≤ i, j ≤ 3). The equation of state for ideal polytropic gas is given by R > 0 is the gas constant. The internal energy flux is defined by the Fourier law as follows: Total energy E T is the sum of internal energy and kinetic energy, given as whereas e = c v θ .
When the director field is a constant vector field, system (1.1) reduces to compressible non-isentropic Navier-Stokes equations. First, let us recall some literature works on the well-posedness theory for the compressible Navier-Stokes equations in a threedimensional space. When the initial density is away from vacuum, Nash [10] and Itaya [11] established the local existence and uniqueness of classical solutions in 1962 and 1977, respectively. In 1980, Matsumura and Nishida [12] first obtained the global classical solutions for the initial data close to a non-vacuum equilibrium in a Sobolev space. When vacuum is allowed, in 2006, Cho and Kim [13] proved the existence of unique local strong solutions in bounded and unbounded domains. In 2018, Huang and Li [14] proved the global existence of classical solutions in the whole space with smooth initial data which are of small energy but possibly large oscillations. For more results on the existence of solutions, see [15][16][17][18][19][20][21][22][23][24] and the references therein.
For liquid crystal in higher dimension, in 2012, Huang et al. [9] constructed the local existence of strong solution with sufficient regular initial data. In the same year, Hu and Wu [25] proved the existence and uniqueness of the global strong solution in critical Besov spaces provided that the initial datum is close to constant equilibrium state (1, 0,d). The spherically symmetric solutions to compressible hydrodynamic flow of liquid crystals can be found in reference [26]. In 2015, Yang [27] obtained global in time strong solution and justified the low Mach number limit in a bounded domain. For the results of compressible liquid crystal flows model with Ginzburg-Landau energy, existence, and large-time behavior of global weak or strong solution, we refer to [28][29][30] and the references therein. For more details about the development in the analysis of liquid crystal, an interested reader can see a very systematic review by Lin and Wang [31]. There are fewer results about nonisothermal nematic liquid crystal equations compared to those on compressible isentropic nematic liquid crystal. In 2017, Guo et al. [32] established the existence of global-in-time smooth solutions in a three-dimensional space provided that the initial datum is close to a steady state. In 2018, the local well-posedness of nematic liquid crystal equations (1.1) was studied by Fan et al. [33] in a bounded domain ⊂ R 3 . In 2019, Francesco and Liu [7] derived a general Ericksen-Leslie system under the action of thermal effects and proved the global-in-time well-posedness of the system for small initial data in the framework of Besov spaces. Recently, Zhong [34] studied the singularity formation of strong solutions to the two-dimensional nematic liquid crystal flows in a bounded domain and proved that the strong solution exists globally if the temporal integral of the maximum norm of the divergence of the velocity is bounded.
All the global well-posedness results obtained for compressible liquid crystal flow in higher dimensional space variable are restricted to smallness assumptions on initial data, energy, and so on, the global existence with smooth initial data in dimension n = 3 is an open problem analogous to Navier-Stokes equations. Although the well-posedness theory for one-dimensional Navier-Stokes is well known, the physical phenomenon of liquid crystal dynamics creates serious difficulties in rigorous analysis. When the director field is a constant vector, the system is reduced to one-dimensional Navier-Stokes equations. For such a particular system of equations, in 1977, Kazhikhov and Shelukhin [35] established the global existence and uniqueness of strong solutions with positive density and constant viscosity coefficient, for the corresponding Cauchy problem, see [36]. Later, asymptotic behavior of the solution was studied by Okada [37] in 1987. Zlotnik and Amosov [38,39] constructed the global weak solutions, and later, Chen, Hoff, and Trivisa [40] the initial and boundary conditions. In 1998, Jiang [41] studied the equations with density-dependent viscosity and proved that if the viscosity does not decrease to zero too rapidly, then smooth solutions exist globally in time. For more related works with density-dependent viscosity, see [42][43][44][45][46]. In 2000, Luo and Xin [47] obtained the regularity and solutions behavior near gas and vacuum interfaces. Large time behavior of solutions with large initial data was recently proved in an unbounded domain by Li and Liang [48]. In 2015, Pan and Zhang [49] proved the existence of global strong solutions when heat conductivity depends on temperature by the following Chapman-Enskog power law: The global well-posedness of solution with β = 0 and vacuum was obtained by Li [50] in 2019, this result holds both for vacuum and non-vacuum case and has improved the well-known result of Kazhikhov [35](non-vacuum case). For the Cauchy problem, with constant coefficients and far field vacuum, Li and Xin [51] obtained the global existence for entropy bounded solution to non-isentropic Navier-Stokes equations with zero heat conduction.
The study of hydrodynamics of liquid crystal has attracted the attention of many researchers for the past few years. For isentropic compressible liquid crystal flow onedimensional space, Ding et al. [52], in 2011, gave the existence of weak solution (ρ, u, d) Ding,Wang,and Wen [53] obtained the existence and uniqueness of global classical solution for Hölder continuous initial data and initial density away from vacuum. In 2015, Huang and Ding [54] studied a free boundary problem for the compressible liquid crystal flow model which connects to vacuum continuously. For non-isothermal compressible liquid crystal flow, in 2019, Tang and Sun [55] proved the global existence of strong solutions allowing vacuum, provided that the initial data satisfy some compatibility condition and the heat conductivity satisfies (1.4) which plays an essential role in obtaining the regularity of the temperature. The result [54] was extended by Mei [56] in 2020, where the author proved global classical solution to the free boundary value problem in the presence of temperature equation with heat conductivity of type (1.4). By using the stronger assumption on heat conductivity (1.3), recently, in 2021, Li, Mahmood, and Shang [57] obtained the global strong solution with ρ 0 ≥ C. To our best knowledge, global strong solution to one-dimensional non-isothermal compressible nematic liquid crystal equations for arbitrary large initial data is not known for constant coefficients and vacuum. The global solution obtained for system (1.5) in Mei [56], Li et al., [57] holds for β > 0 with ρ 0 ≥ C. Similarly, the result obtained in [55] with vacuum in Euler coordinates is restricted to the condition on heat conductivity of type (1.4). Motivated by Kazhikhov [35] and Li [50], in this paper we aim to study the global well-posedness of strong solutions to the one-dimensional non-isothermal compressible nematic liquid crystal flow equations, i.e., system (1.5), with constant viscosity and heat conductivity in the presence of vacuum.

Coordinates transformation and main results
Let ρ(x, t), u(x, t), d(x, t), and θ (x, t) denote the unknown functions of density, velocity, director vector field(d(x, t) = (d 1 , d 2 , d 2 )(x, t)), and temperature, then the liquid crystal hydrodynamics for all x ∈ (0, l) and time t > 0 is governed by the following set of partial differential equations in Eulerian coordinates: where μ and κ are viscosity and heat conductivity coefficients and μ = 2μ + λ .
The results of this paper will be proven in the Lagrangian flow map coordinate be- ing stated in what follows. Let us define the coordinate transformation between the Lagrangian coordinate y and Euler coordinates x as where ξ (y, t) denotes the flow map governed by u, that is, ξ (y, 0) = y.
Let the new variablesρ,ū,dp,θ denote the density, velocity, director field, pressure, and temperature, respectively, in the Lagrangian coordinate, that is, By the definition of ξ (y, t), it is not hard to see that and introduce a function g(y, t) = ξ y (y, t). Then it follows For the sake of simplicity, new variables are denoted by the same physical variables as before, then system (1.5) can be rewritten in the Lagrangian coordinate as follows: (1.7) Due to (1.6) and (1.7) 1 , it is straightforward that from which, by setting ρ| t=0 = ρ 0 and noticing that g| t=0 = 1, we have gρ = ρ 0 . Therefore, one can replace ρ by ρ = ρ 0 g in the above equations, and a new modified system is written as follows: For 1 ≤ r ≤ ∞ and positive integer k, we use L r = L r ((0, l)) and W k,r = W k,r ((0, l)) to denote the standard Lebesgue and Sobolev spaces, respectively, and in the case that r = 2, we use H k instead of W k,2 . H 1 0 consists of all functions v ∈ H 1 satisfying u(0) = u(l) = 0. We always use u r to denote the L r norm of u. Throughout this paper, C denotes a general nonnegative constant which may be different from line to line.
The main result of this paper is stated as follows.
Then there is a unique global solution (g, u, d, θ ) to system (1.8) subject to initial-boundary conditions (1.9-1.10) satisfying for any T ∈ (0, ∞). In particular, it holds that for any T > 0, where C > 0 is a positive constant depending only on R, c v , μ, κ, m 1 , N 1 , N 2 , N 3 , and T, and dissipation D(t) is defined as Remark 1 The result of this paper can be seen as a generalization of Kazhikhov-Shelukhin [35] (constant coefficients, non-vacuum) and Li [50] (constant coefficients and vacuum) towards liquid crystal dynamics.
Remark 2 The arguments of this paper also work for the free boundary value problem in which the boundary condition for velocity u is replaced by Thus our result improves the results of [56,57] for κ(θ ) = θ β with β = 0, which does not include the constant case.
Remark 3 The same result as in Theorem 1.1 still holds if we replace the boundary condition ∂ y θ (0, t) = ∂ y θ (l, t) = 0 by one of the following three: and the proof is exactly the same as the one established in this paper, the only difference is that the basic energy identity in Lemma 1 will be replaced by an inequality.
Remark 4 The method used in this paper can be applied to another model, e.g., magnetohydrodynamics, by slightly modifying the proof of this paper.
The key issue of proving Theorem 1.1 is to establish the appropriate a priori energy estimates for the solution of (1.8) up to any finite time, subject to initial and boundary conditions (1.9)-(1.10). Compared to Navier-Stokes, the presence of highly nonlinear director field equation makes the analysis more difficult due to production of g| 1 g ( d y g ) y + 1 g 2 |d y | 2 d| 2 in the temperature equation, the term |d y | 2 g 2 d, constraint |d| 2 = 1 in the director field equation, and coupling of the torque balance equation with the momentum balance equation with the term 1 g 2 |d y | 2 . In addition to the difficulty caused by the lack of the positive lower bound of the density, it is observed that upper bound density strongly depends on the L ∞ -norm of d y g and the quantity ρ 0 θ . Strong coupling of cross terms and vacuum causes serious difficulties in the control of L ∞ (L 2 ) of √ ρ 0 θ . To overcome such difficulties, densityweighted embedding inequalities are used both for temperature and the director field. The desired a priori energy estimates are carried out as, first, we obtain from (1.8) an identity as follows: for some functions X(t), Y (y, t), and Z(y, t) = exp(-t 0 1 2g 2 |d y | 2 ds). The temperature equation is not used at all in deriving the above identity, and this identity is in the spirit of the one in [35], but in different Lagrangian coordinates. The basic energy estimate implies that both X and Y are uniformly away from zero and uniformly bounded up to any finite time. As a direct corollary of the above identity, one can obtain the uniform positive lower bound of g and the control of the upper bound of g in terms of t 0 θ ds and t 0 1 2g 2 |d y | 2 ds. By using the positive lower bound of g, we obtain a density-weighted embedding inequality which implies that the upper bound of √ ρ 0 θ and 1 √ ρ 0 | d y g | 2 can be controlled by that of g, up to a small dependence on This will be used for the L ∞ (L 2 ) type estimates on √ ρ 0 θ .
Second, we carry out the L ∞ (L 2 ) energy estimate on √ ρ 0 θ and, at the same time, the L ∞ (L 2 ) energy estimate will be evolved naturally, due to the coupling structure between u, d, and θ in the system. Compared to Navier-Stokes equations, we do not have control for L ∞ (L 2 ) energy estimate on √ ρ 0 θ because of time integral term of the director field in product with time integral of temperature. It is noted that there is no dissipation estimates on the director field equation due to energy conservation. As a conclusion, by extracting dissipation estimates from momentum and torque balance equation in terms of θ ∞ , we are able to obtain the a priori upper bound of g and the a priori L ∞ (L 2 ) ∩ L 2 (H 1 ) type estimates on (u, d y g , θ ). Third, by using the effective viscous flux F := μ ∂ y u gp - 1 2 | d y g | 2 and working on its L ∞ (L 2 ) ∩ L 2 (H 1 ) type a priori estimate that strongly depends on the term d yt √ g 2 L 2 of the director field, we are able to get the a priori L ∞ (H 1 ) estimate on (g, u); however, due to the presence of the term μ g (∂ y u) 2 and g|d t | 2 and the degeneracy of the leading term ρ 0 ∂ t θ in the θ equation, we are not able to obtain the corresponding L ∞ (H 1 ) estimate on θ without appealing to higher order energy estimates than H 1 .
Lastly, we obtained the a priori L ∞ (H 2 ) type estimates on (u, θ ), which are achieved through performing the L ∞ (L 2 ) type energy estimate on √ ρ 0 ∂ t θ , L ∞ (H 1 ) type estimate on F, and by using a priori L ∞ (H 3 ) estimates on the director field. It should be mentioned that the desired a priori L ∞ (H 2 ) estimates on θ are obtained without knowing their a priori L ∞ (H 1 ) bound in advance.
In fact, all the energy estimates obtained in this paper hold if we replace boundary condition (1.10) with the above one by copying or slightly modifying the proof.

A priori L 2 estimates
We start with the basic energy identity.

the conservation of mass and energy is given as
Proof Integrating (1.8) 1 with respect to space and time and using boundary condition (1.10), it is easy to see that equality (2.1) holds. In order to prove (2.2), we multiply (1.8) 2 by u. The resulting equation is then integrated over [0,l]; after integration by parts, we obtain On the other hand, taking dot product of equation ( , as a result one can get The left-hand side of (2.4) can be written as where we have used constraint |d| 2 = 1. Plugging (2.5) into (2.4) and then multiplying the resulting equation by g, we get (2.6) Then, we integrate equality (2.6) over [0,l], and we have Combining (2.3) and (2.7) with integral form of the temperature equation (1.8) 4 implies equality (2.2).
Next, we carry out the estimate on the lower bound of g. To this end, we perform some calculations in the spirit of [35] as preparations. Now, we integrate momentum conservation equation with respect to time t, and using first equation of (1.8), we can get Integrating (2.8) with respect to y from the point ξ (t) to an arbitrary point y for any fixed time t, we get After rewriting, we have where Multiplying equation ( From (2.10) and (2.11) we get A prior positive lower and upper bound of g is stated in the following proposition.
Proof By Lemma 1, it follows from the Hölder inequality that Therefore, it follows from the definition of Y in Lemma 1 that Similarly, Now we show that X(t) is bounded above and below by the initial data. Assume that then With the aid of the continuity equation, we obtain Integrating it over [0, l] × [0, t], using non-slip boundary (1.10) condition, we get By using mass conservation (2.1), we have Using the continuity of g, On the other hand, since u y = g t , then from the definition of F we have By rewriting in the form Thus it is not hard to see that Recalling p ≥ 0, we obtain the conclusion follows.
Next we establish the upper bound of g, where we need to prove ρ 0 θ L ∞ and for the director field ( ∞ type a priori estimate. The need of the weighted estimate for temperature and director field is necessary for the upper bound of g. As a preparation of deriving the a priori upper bound of g and the a priori L ∞ (0, T; L 2 ) ∩ L 2 (0, T; H 1 ) type estimates on (u, d y , θ ), for simplicity, the density-weighted estimates of θ and d y are given in the following proposition.
Noticing that 1 2 , thus one can get Density-weighted estimate for the director field: By the Hölder and Young inequalities, and similarly as above, we deduce for any y ∈ (0, l). Thus, from Lemma 1 and suitably small δ > 0, we have for a positive constant C depending only on N 1 . Therefore, we have Density-weighted estimate for temperature: Assume ρ ∞ =ρ, let there exist y 0 ∈ (0, l) such that By the Hölder and Young inequalities, we deduce for any y ∈ (0, l), it is not hard to see that Noticing that we deduce, by the Hölder inequality, that The other estimates for θ in Proposition 2 can be obtained similar to the director field. By density-weighted estimate of θ and d y in hand, the desired a priori L ∞ (0, T; L 2 ) ∩ L 2 (0, T; H 1 ) estimates on (u, d y g , θ ) are given as in the following lemma.
g 2 ) + c v θ and integrating the resultant over (0, l), one gets from integration by parts that  dy, for a positive constant C depending only on R, c v , μ, and κ. Substituting the above two inequalities into (2.21) and applying Lemma 1 and Proposition 1, we obtain d dt for a positive constant C depending only on R, c v , μ, and κ, and thus d dt for a positive constant C 1 depending only on R, c v , μ, and κ. Multiplying the momentum equation by 4u 3 and integrating the resultant over (0, l), after integration by parts and the Young inequality, one gets that , from which, by Proposition 1, the Hölder inequality, and Proposition 1, one obtains Now we first operate the ∂ y on both sides of the director field equation Taking the dot product of the equation by ( Now we compute each term on the left-and right-hand side of equation ( In the above calculation we have used the boundary conditions d y = 0, u = 0. For the terms on the right-hand side, we calculate them one by one. The first term after integration by parts and using boundary condition is as follows: for any t ∈ (0, T).
By Proposition 1 and (ii) of Proposition 2, we have Collecting the estimates provides the proof of the lemma.
In order to close the estimates of the lemma, we need to control the unbounded operator We see that the director field term is in product with the time integral of L ∞norm of temperature, so it cannot be controlled by right-hand side terms, thus we need new estimates to control these terms. In order to control the right-hand side of the above inequality, dissipation estimates are obtained on the director field in terms of L ∞norm of temperature.
Proof Multiplying (1.8) 2 by u and taking the dot product of director field equation (1.8) 3 with -( 1 g ( d y g ) y + |d y | 2 d g 2 ) respectively, the resultant is integrated over (0, l). From the momentum equation it is not hard to see that The constraint |d| 2 = 1 is used in the above inequality, thus, by Proposition 1, Thus we see that, in order to close the above estimates, we need a control on temperature, which will be given in the following lemma.
Control of term θ ∞ : Now we prove the following key lemma.

Lemma 4 For any t
whereas C depends on ρ 0 ∞ and the initial data.
Proof Multiplying the temperature equation by (θ + δ) -1 for some δ ∈ (0, 1), the resultant equation can be written as Integrating the above inequality over time and space variables for (θ + δ) ≥ 1, then using boundary condition for temperature, we get Thus we get From Proposition 2, inequality (i) 4 , it holds that This completes Lemma 4.
Combining all these lemmas, we have the following proposition.

Proposition 3 Given T ∈ (0, ∞). It holds that
for a positive constant C depending only on the initial data R, c v , μ, κ, m 1 , N 1 , and T, where m 1 and N 1 are the numbers in Proposition 1 and Proposition 2, respectively.
Proof Summing with (2.28), one obtains d dt for any t ∈ (0, T), where C is a positive constant depending only on R, c v , μ, κ, m 1 , N 1 , and T. Applying the Gronwall inequality to the above inequality, one gets

A priori H 1 estimates
This section is devoted to the a priori H 1 type estimates on (g, u, d y , θ ). Precisely, we will carry out the a priori L ∞ (0, T; H 1 ) ∩ L 2 (0, T; H 2 ) estimate on u, a priori L ∞ (0, T; H 1 ) ∩ L 2 (0, T; H 2 ) estimate on d y , and the L ∞ (0, T; H 1 ) estimate on g; however, due to the presence of the term μ g (∂ y u) 2 and g| 1 g ( d y g ) y + 1 g 2 |d y | 2 d| 2 on the right-hand side of the equation for θ (1.8) 4 , one cannot get the desired a priori H 1 estimate of θ independent of the lower bound of the density without appealing to the higher than H 1 energy estimates. Before going to prove the H 1 bound for velocity, we first give the following estimates of director field, because the velocity field estimates strongly depend on the director field. The term on the left-hand side of (2.43) can be written with the aid of (1.8) 1 and constraints |d| 2 = 1 as follows:

Lemma 5 There exists a positive constant C > 0 such that, for any t
Next the estimate of each term is given as follows: The term Q 1 is estimated as Now the estimate on the last term Q 3 is given as (2.48) By using the Hölder inequality and embedding W 1,1 (0, l) → L ∞ (0, l), Q 2 is estimated as whereas in the fourth inequality the following fact is used: 1 g ( d y g ) y · d + |d y | 2 g 2 = 0. Plugging (2.50) into (2.49), using Young's inequality, we get where we have used the fact that Then, one can derive from (1.8) that Moreover, by equation (1.8) 2 , one has ∂ y F = ρ 0 ∂ t u, from which, recalling the boundary conditions, we have We have the a priori L 2 estimates on F stated in the following.

Proposition 4
Given T ∈ (0, ∞). It holds that Using ( Therefore, it follows from the Hölder, Young, and Gagliardo-Nirenberg inequalities and Corollary 3 that Now we estimate each term as follows: for any t ∈ (0, T), where C is a positive constant depending only on R, c v , μ, κ, m 1 . Applying the Gronwall inequality to (2.58) and using Corollary 3, the conclusion follows.
Based on Proposition 4 and Corollary 3, we can obtain the desired H 1 type estimates on g, u, 1 g ( d y g ) y , stated as follows.
The estimates of velocity term in I 1 and I 2 are straightforward from a priori estimates in hand and by the definition of F, noticing that ρ 0 ∂ t u = ∂ y F. Note that, by the Sobolev embedding inequality, it follows from Proposition 4 that for a positive constant C depending only on R, c v , μ, κ, m 1 , N 1 , N 2 , and T. However, the a priori estimates on terms ∂ y g 2 2 and T 0 ∂ 2 y u 2 2 dt need to be computed. Rewrite (1.8) 1 in terms of F as Differentiating the above equations in y, multiplying the resultant by ∂ y g, and integrating over (0, l), it follows from the Hölder and Young inequalities that 1 μ l 0 F|∂ y g| 2 + ∂ y Fg∂ y g + R ρ 0 θ + ρ 0 ∂ y θ ∂ y g dy |∂ y g| 2 dy =: T .

Now we compute T as
for a positive constant C depending only on R and μ. Applying the Gronwall inequality, it follows from (2.60), Corollary 3, and Proposition 4 that for a positive constant C depending only on R, c v , μ, κ, m 1 , N 1 , N 2 , and T.
We summarize the estimates obtained in this subsection as follows.

A priori H 2 estimates
This subsection is devoted to the a prior H 2 estimates on (g, u, d y , θ ). As will be shown in this subsection, one can get the desired a priori L ∞ (0, T; H 2 ) estimate of θ without using the a priori L ∞ (0, T; H 1 ) bound of it. As a preparation, we first give some estimates on ∂ y θ 2 and ∂ t θ ∞ in terms of √ ρ 0 ∂ t θ 2 and ∂ y ∂ t θ 2 , and ∂ t u ∞ , in terms of √ ρ 0 ∂ t u 2 and ∂ y ∂ t u 2 , which will be used later in higher order a priori estimates.
(ii) It holds that where 0 is the number in Proposition 2.
Proof (i) Multiplying (2.86) by θ , integrating the resultant over (0, l), and integrating by parts, it follows from the Hölder inequality that from which, by Corollaries 3-1 and (i) of Proposition 2, we have that and thus, for a positive constant C depending only on R, c v , μ, κ, m 1 , N 1 , N 2 , and T.
Proposition 7 Given T ∈ (0, ∞). It holds that for a positive constant C depending only on R, c v , μ, κ, m 1 , N 1 , N 2 , and T, where or equivalently, from which, differentiating in t and using (1.8) 1 , one has Multiplying the above equation by ∂ t θ , integrating the resultant over (0, l), one gets from integration by parts that The terms on the right-hand side of (2.64) are estimated as follows. By Corollary 3, it follows from the Young inequality and (i) of Proposition 6 that for a positive constant C depending only on R, c v , μ, κ, m 1 , N 1 , and T. By Corollary 3, Corollary 1, and (ii) of Proposition 6, it follows from the Hölder and Young inequalities that 1 μ l 0 2gF + Rρ 0 θ + 1 2 for a positive constant C depending only on R, c v , μ, κ, m 1 , N 1 , N 2 , and T. Before going to estimate the next term, we first compute The right-hand side is estimated as follows: The second integral on the right-hand side of (2.64) is estimated as where we have used the Hölder, Young inequalities and mass conservation equation. Therefore, one obtains from (2.64) that for a positive constant C 3 depending only on R, c v , μ, κ, m 1 , N 1 , N 2 , and T. In order to control the above estimates, there is a need to get dissipation estimates on ( √ gd tt 2 , √ g∂ t F 2 2 ). Next, we give the higher order derivative estimates of the director vector field d. The identity 1 g ( d y g ) y · d + |d y | 2 g 2 = 0, which is obtained from the director field equation with the help of constraint |d| 2 = 1, is frequently used in the proof of the lemma. This fact replaces the higher derivative with nonlinearity. Proof In order to prove higher order derivative estimates on the director field, the derivative with respect to y variable is taken on both sides of director field equation (1.8) 3 and the resultant is multiplied by 1 g , then it follows (2.70) Taking the dot product of (2.70) on both sides with ( 1 g ( d y g ) y ) yt and integrating over space variable, it follows that where in the second equality we have used 1 g ( d y g ) y ·d + |d y | 2 g 2 = 0. This fact replaces the higher derivative with nonlinearity. The key point is to control the higher derivative with lower one by using the constraint |d| 2 = 1. Now the terms R 1 , R 2 , R 3 are estimated one by one.