Abstract
In this paper, for the incompressible nematic liquid crystal flows with heat effect and density-dependent viscosity coefficient in three-dimensional bounded domain, by using the elliptic regularity result of the Stokes equations and the linearization and iteration method, we investigate the local existence and uniqueness of strong solutions.
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1 Introduction
Let \(\Omega \subset {\mathbb {R}}^3\) be a bounded domain with smooth boundary \(\partial \Omega \) whose unit outward normal is \(\vec {n}\), we are concerned with the incompressible nematic liquid crystal flows with heat effect and density-dependent viscosity coefficient in \(\Omega \):
where \(\rho \), u, \(\theta \) and p are fluid density, velocity, absolute temperature and the pressure, respectively. The constants \(\lambda >0\), \(c_{\nu }>0\) and \(\vartheta >0\) are competition between kinetic and potential energy, heat conductivity and microscopic elastic relaxation time, respectively; for convenience, we will set \(\lambda =\vartheta =1\) in this paper. The force term \(\nabla d\odot \nabla d\) in the equation of the conservation of momentum denotes the \(3\times 3\) matrix whose ijth entry is given by ‘\(\nabla _id\cdot \nabla _jd\)’ for \(1\le i,j\le 3\). We also denote by \({\mathcal {D}}(u)\) the deformation tensor \(\frac{1}{2}[\nabla u+(\nabla u)^T]\). The two continuously differentiable functions \(\mu (\rho )\) and \(\kappa (\rho )\) satisfy
for some positive constants \(\underline{\mu }\) and \(\underline{\kappa }\). System (1.1) is a simplified version of those proposed in [49, Chapter BIII2], [51, Chapter 3], [40, Section 9] and [21, Section 2].
In this paper, on the basis of physical considerations, we suppose that equations (1.1) is coupled with the following initial conditions
and boundary value conditions
where \(\vec {n}\) is the unit outward normal vector to \(\partial \Omega \).
If \(\rho \equiv \hbox {constant}>0\) and the heat effect is neglected, the simplified incompressible nematic liquid crystals flow is [12, 13, 35]
In order to understand the nematic liquid crystal flows distinctly, Lin [41] replaced the nonlinear term \(|\nabla d|^2d\) with \(|d|=1\) by the Ginzburg–Landau-type approximation term \(\frac{1}{\eta ^2}(|d|^2-1)d\). Since then, there has been a lot of literatures studied the global well-posedness and regularity criterion for simplified Ericksen–Leslie system with Ginzburg–Landau approximation (see, for example, Calderer, Golovaty, Lin and Liu [3], Lin and Liu[43], Lin, Lin and Wang [42], Fan, Alzahrani, Hayat, Nakamura and Zhou [14] and the reference cited therein). For the case of \(|\nabla d|^2d\) with \(|d|=1\), there are also many researches involves (cf. [7, 15, 25, 26, 53, 61] and the references therein).
If \(\rho \equiv \hbox {constant}>0\) and the heat effect is considered, system (1.1) seems more complicated. Feireisl and his coauthors [20, 21] studied the approximated system and established the existence of the global weak solutions in 2D and 3D cases. Li and Xin [40] also considered the existence of global weak solutions to the nematic liquid crystal flows with heat effect in \({\mathbb {R}}^2\). Recently, letting \(\rho \equiv \hbox {constant}>0\), the continuously differentiable function \(\mu =\mu (\theta )\) and \(\kappa \) is a positive constant, Bian and Xiao [1] have obtained the following temperature-dependent incompressible nematic liquid crystal flows in a bounded domain \(\Omega \subset {\mathbb {R}}^n\) (\(n=2,3\)):
Following Danchin’s method [8], the authors investigated the small initial data global existence and uniqueness of strong solutions.
Mathematically, system (1.1) is made up of the density-dependent incompressible Navier–Stokes equations, the transported heat flows of harmonic map and the internal energy equation describing the evolution of temperature. There are a lot of classical results on the existence of weak solutions to the density-dependent incompressible Navier–Stokes equations due to the physical importance, complexity, rich phenomena and mathematical challenges. When the viscosity \(\mu \) is a positive constant and \(\inf \rho _0>0\), Kazhikhov [32] studied the existence of global weak solutions with finite energy. Later, Simon removed the lower bound assumption on \(\rho _0\) to solve the global weak solution; Lions [44] proved that \(\rho \) is a renormalized solution of the mass equation, which enable him to consider the global weak solutions for 3D inhomogeneous Navier–Stokes equations with the viscosity \(\mu =\mu (\rho )\). Yet the uniqueness and regularities of such weak solutions are big open questions even in two space dimension [44]. The study on strong solutions for inhomogeneous Navier–Stokes equations can be trace back to Ladyzhenskaya and Solonnikov [33]. The authors constructed the global strong solution for 2D equations in the bounded domain case, whenever the initial data \(u_0\in H_0^1\) and \(\rho _0\in W^{1,\infty }\) with \(\inf \rho _0>0\). Henceforth, a number of papers have been devoted to the study of strong solutions to inhomogeneous Navier–Stokes equations (cf. [8, 9, 28, 30, 50, 54, 57] and the reference therein). Recently, for the initial density allowing vacuum, Cho and Kim [5] have established the local existence of unique strong solutions of density-dependent incompressible Navier–Stokes equations for initial data satisfying a natural compatibility condition and the bounded domain \(\Omega \subset {\mathbb {R}}^3\) (see also [6] for the density-dependent heat conducting Navier–Stokes equations). It is worth pointing out that Li [37] removed the compatibility condition, established the local well-posedness result for the inhomogeneous Navier–Stokes equation with vacuum and \(\mu \) is a positive constant and the non-negative initial density \(\rho \in W^{1,q}(\Omega )\), where \(q\in (3,\infty )\); Danchin and Mucha [10] dropped the restrict on the derivative of initial density, only assume that \(0\le \rho _0\le C\) and \(\int _{\Omega }\rho _0dx>0\), obtained the global unique solutions in 2D and small initial velocity global unique solutions in 3D. However, the generalization to the case \(\mu =\mu (\rho )\) depends on \(\rho \) within Li’s approach or Danchin–Mucha’s approach being unclear as regards local-in-time and global-in-time results. On the other hand, motivated by [39, 46] dealing with the compressible flow, Lv, Xu and Zhong [48] established the local existence of strong solutions to the Cauchy problem of the 2D density-dependent MHD equations with vacuum (the MHD equations reduces to the Navier–Stokes equations as the magnetic field \(B = 0\)). For the investigations on the compressible Navier–Stokes equations, we refer the reader to [2, 4, 19, 27, 29, 47] and the reference therein.
There are also some results on the density-dependent incompressible nematic liquid flows with \(\mu \equiv \hbox {constant}>0\) (see [16,17,18, 24, 36, 38, 56]). Moreover, for the 3D non-homogeneous incompressible nematic liquid flows with the viscosity \(\mu (\rho )\) depends on the density \(\rho \), we recall that Gao, Tao and Yao [23] studied the local well-posedness of strong solutions, Yu and Zhang [55] and Liu [45] established the existence of small initial data global-in-time strong solutions when the initial vacuum is allowed. If the heat effect is considered, the system is energetically closed but more complicated. So far, we only find some papers related to the density-dependent incompressible heat-conducting Navier–Stokes equations (i.e. system (1.1) with \(d=0\)) [6, 58,59,60]. However, there is no other result on the mathematical analysis of system (1.1), and the local existence and uniqueness of strong solutions to system (1.1) with vacuum are still unknown. In fact, this is the main aim of this paper. Assume that the initial data satisfy a compatibility condition (see (1.7)), then by using the elliptic regularity result of the Stokes equations and the linearization and iteration method, we prove the local existence and uniqueness of strong solutions to system (1.1). Our main results read as follows:
Theorem 1.1
Let \(q\in (3,\infty )\) be a fixed constant and the initial data \((\rho _0,u_0,\theta _0)\) satisfy the regularity conditions
and the compatibility conditions
for some \(p_0\in H^1\) and \(g_1,~g_2\in L^2\). Then, there exist a positive time \(T>0\) and a unique strong solution \((\rho ,u,\theta ,p)\) for problem (1.1)–(1.4) such that
for some \(r\in (3,\min \{6,q\})\).
Remark 1.2
Since the uniqueness of the solutions to problem (1.1)–(1.4) is easy to obtain by a standard argument [44], we only consider the existence result in Theorem 1.1. There are three parts for us to prove: In part one, we establish the global strong solutions for some linearized system related to problem (1.1)–(1.4). Part two is devoted to prove the solutions of the linearized system converges to the original initial problem in a local time for positive initial density. In the end, we verify Theorem 1.1 for the case of general initial density with vacuum.
Remark 1.3
The blowup criteria and the global existence of strong solution for problem (1.1)–(1.4) will be the subject of the forthcoming papers. Moreover, we will also improve this result to degenerated heat effect case in the following papers.
Remark 1.4
For system (1.1), the equations of \((\rho , u,d)\) do not depend on the temperature \(\theta \). Therefore, the system of \((\rho , u,d)\) is closed. In this case, Gao, Tao and Yao [23] studied the existence and uniqueness of local strong solutions. Since the needed a priori estimates on \((\rho ,u,d)\) was established in [23], for simplicity, we omit their proof, only list them in this paper. The main difference between [23] and this paper is the heat effect is considered in this paper. Thanks to the heat effect, the system is energetically closed but more complicated. In order to study the local well-posedness, we have to pay much attention on the a priori estimates of heat effect. On the other hand, compared with the density-dependent incompressible Navier–Stokes equations with heat effect [6, 58,59,60], there exists a nonlinear strong term \(\rho |\Delta d+|\nabla d|^2d|^2\) for Eq. (1.1)\(_4\). In order to obtain the local well-posedness of system (1.1), we have to pay a lot of pages to deal with this term and obtain suitable a priori estimates.
The paper is written in the following way: in Sect. 2, we give some notations and basic lemmas. Then, in Sect. 3, we prove the existence and uniqueness of strong solutions to a linearized equations related to problem (1.1)–(1.4). The proof of Theorem 1.1 is postponed in Sect. 4.
2 Preliminaries
For simplicity, we denote
We begin with the following regularity results of density-dependent Stokes equations.
Lemma 2.1
([5]) Let \(\mu (\rho )\) satisfies (1.2) with \(\mu (\rho )\in W^{1,p}\) and \(0\le \rho \le \overline{\rho }\) for some positive number \(p\in (3,\infty )\). Assume that \((u,p)\in H^1_{0 }\times L^2\) is the unique weak solution to the following boundary value problem
Then the following regularity estimates hold for (u, p):
-
if \(F\in L^2\), then \((u,p)\in H^2\times H^1\) and
$$\begin{aligned} \Vert u\Vert _{H^2}+\Vert p\Vert _{H^1}\le C\Vert F\Vert _{L^2}(1+\Vert \nabla \mu (\rho )\Vert _{L^p})^{\frac{p}{p-3}}. \end{aligned}$$ -
If \(F\in L^r\) for some \(r\in (n,p)\), then \((u,p)\in W^{2,r}\times W^{1,r}\) and
$$\begin{aligned} \Vert u\Vert _{W^{2,r}}+\Vert p\Vert _{W^{1,r}}\le C\Vert F\Vert _{L^r}(1+\Vert \nabla \mu (\rho )\Vert _{L^p})^{\frac{pr}{2p-2r}}. \end{aligned}$$
Here, the positive constant C depends only on r, p, \(\Omega \), \(\underline{\mu }\), \(\overline{\mu }\) and \(\overline{\rho }\).
Next, the following well-known Gronwall’s Lemma will be frequently used later.
Lemma 2.2
(Gronwall Lemma [52]) Let g, h, y and \(\frac{dy}{dt}\) be locally integrable functions on \((t_0,\infty )\) such that
then y(t) satisfies
We introduce the Poincaré inequality in Sobolev space.
Lemma 2.3
(Poincaré inequality [11]) Suppose that \(\Omega \subset {\mathbb {R}}^n\), \(n\in {\mathbb {N}}^+\), then
-
if \(u\in H_0^1(\Omega )\), we have
$$\begin{aligned} \Vert u\Vert _{L^2}\le \left\{ \begin{aligned}&\frac{|\Omega |}{\pi }\Vert \nabla u\Vert _{L^2},\quad \quad N=1, \\&C(\Omega )\Vert \nabla u\Vert _{L^2},\quad N\ge 2, \end{aligned}\right. \end{aligned}$$(2.2) -
if \(u\in H^1(\Omega )\), we have
$$\begin{aligned} \Vert u\Vert _{L^2}^2\le \left\{ \begin{aligned}&\frac{|\Omega |^2}{2}\Vert \nabla u\Vert _{L^2}^2+\frac{1}{|\Omega |} \left( \int _{\Omega }udx\right) ^2,\quad \quad N=1, \\&C(\Omega )\left\{ \Vert \nabla u\Vert _{L^2}^2+\left( \int _{\Omega }udx\right) ^2\right\} , \quad \quad N\ge 2. \end{aligned}\right. \end{aligned}$$(2.3)
We also introduce the Gagliardo–Nirenberg inequality:
Lemma 2.4
(Gagliardo–Nirenberg inequality [22]) Let u belongs to \(L^q({\mathbb {R}}^n)\) and its derivatives of order m, \(\Lambda ^m u\), belong to \(L^{r}\), \(1\le q,r\le \infty \). For the derivatives \(\Lambda ^j u\), \(0\le j<m\), the following inequalities hold
where
for all \(\alpha \) in the interval
(the constant depending only on \(n,m,j,q,r,\alpha \)), with the following exceptional cases
-
1.
If \(j=0\), \(rm<n\), \(q=\infty \), then we make the additional assumption that either u tends to zero at infinity or \(u\in L_{\tilde{q}}\) for some finite \(\tilde{q}>0\).
-
2.
If \(1<r<\infty \), and \(m-j-n/r \) is a non-negative integer, then (2.4) holds only for a satisfying \(j/m\le \alpha <1\).
3 Global Existence of the Linearized Equations
In order to prove Theorem 1.1, we first construct the approximate solutions to the initial boundary value problem (1.1)–(1.4) by solving iteratively the following linearized problem:
where v is a known divergence-free vector field, \(2{\mathcal {D}}(u)=\nabla u+\nabla ^Tu\), \(\mu = \mu (\rho )\) and \(\kappa =\kappa (\rho )\) satisfy (1.2).
For problem (3.1), we state the following result.
Lemma 3.1
Suppose that \(q\in (3,\infty )\) is a fixed constant and the initial data \((\rho _0,u_0,\theta _0)\) satisfy the regularity conditions
and the compatibility conditions
for some \(p_0\in H^1\) and \(g_1,~g_2\in L^2\). Moreover, v satisfies the regularity conditions
for some \(r\in (3,\min \{q,6\})\). Then, there exists a unique strong solution \((\rho ,u,\theta ,p)\) to system (3.1) such that
Remark 3.2
The similar results on the coupled system of linear transport equation and the non-stationary Stokes equations were studied by Cho and Kim [5] and Gao, Tao and Yao [23]. Compared with [5], our advantage is less constraint on the viscosity coefficient \(\mu =\mu (\rho )\). In addition, in order to deal with the density-dependent heat conducting nematic liquid crystal flows, we have to add two compatibility conditions on the initial data \(u_0\) and \(\theta _0\). The analysis and calculation are more complicated than Gao, Tao and Yao [23].
Proof of Lemma 3.1
We first analysis the existence and regularity of a unique strong solution to the linear transport equation (3.1)\(_1\). By means of the classical method of characteristics [31, 33], we obtain
Taking the gradient operator \(\nabla \) to (3.1)\(_1\), multiplying by \(|\nabla \rho |^{q-2}\nabla \rho \), integrating by parts over \(\Omega \), it yields that
we obtain \(\rho _t\in L^{\infty }(0,T;L^q)\).
For (3.1)\(_2\)–(3.1)\(_3\), since the uniqueness of strong solutions can be easily proved, we will focus on the proof of existence by using a standard semi-discrete Galerkin method. Define
Suppose that \(\varphi ^m\) is the mth eigenfunction of the Stokes operator \(A=-{\mathcal {P}}\Delta \) in X, where \({\mathcal {P}}\) is the usual projection operator on the divergence-free vector fields. Define
Note that (see [34])
Define
Suppose that \(\phi ^m\) is the mth eigenfunction of the operator \(-\Delta \) in Y, define
we have
Fix a constant \(\delta \in (0,1)\). Let \(\rho ^{\delta }=\rho +\delta \) and \((u_0^{\delta },\theta _0^{\delta },p_0^{\delta })\in H_0^1\times H^1\times L^2\) be a solution to
It then follows from [5] that \((u_0^{\delta },\theta _0^{\delta },p_0^{\delta })\in H^2\times H^2\times H^1\) and
Using standard Galerkin method, we can construct an approximate solution
such that for all \(\omega \in X^m\),
and for all \(\varpi \in Y^m\),
where \(\mu ^{\delta }=\mu (\rho ^{\delta })\), \(\kappa ^{\delta }=\kappa (\rho ^{\delta })\). In fact, we have \(\rho ^{\delta }\ge \delta >0\) in \(\Omega \). Using the regularity of \(\rho ^{\delta }\), we rewrite equations (3.6) and (3.7) with \(\omega =\varphi ^k\) and \(\varpi =\phi ^k\) (\(k=1,2,\cdots ,m\)) as linear ODE systems, respectively. Hence, the existence of a unique Galerkin solution \((u^m,\theta ^m)\) follows from the theory of linear ODEs with regular coefficients.
We also need to establish some uniform bounds to show that a subsequence of the approximate solutions \((u^m,\theta ^m)\) converges to a solution of the original problem (3.1). It follows from (1.2) and (3.4) that
Throughout the proof of this lemma, C denotes a generic positive constant independent of \(\delta \), \(\Vert \rho _0\Vert _{W^{2,q}}\), \(\Vert \partial ^2\mu /\partial \rho ^2\Vert _C\) and \(\Vert \partial ^2\kappa /\partial \rho ^2\Vert _C\).
By using the same method as [5, 23], we easily obtain the following estimates:
and
Taking \(\varpi =1\) in (3.7), it yields that
For any fixed \(\tau \in (0,t)\), integrating (3.11) over \((\tau ,t)\subset [0,T]\), we arrive at
Letting \(\tau \rightarrow 0^+\), it means
Denote \(\overline{\theta ^m}=\frac{1}{|\Omega |}\int \theta ^mdx\). Using Poincaré’s inequality, we obtain
Hence,
Taking \(\varpi =\theta _t^m\) in (3.7), we arrive at
Owning to (3.10), we deduce that
Therefore,
By virtue of Hölder’s inequality, we obtain
Since
then integration by parts gives
Moreover, we also have
Combining (3.15)–(3.19) together gives
By virtue of (3.1)\(_3\), we derive that
which implies that
Combining (3.20) and (3.21) together gives
On the other hand, we have
Hence, for (3.22), by using Gronwall’s inequality, we derive that
Owning to (3.14) and (3.23), we obtain
Differentiating (3.7) with respect to t, taking \(\varpi =\theta _t^m\), it yields that
Applying (3.21), Sobolev’s inequality and Hölder’s inequality, we find that
and
Substituting (3.26)–(3.32) into (3.25) yields
By using Gronwall’s inequality, we arrive at
Since \(\rho ^{\delta }\ge \delta >0\), \(u^m(0)=\sum _{k=1}^m (u_0^{\delta },\varphi ^k)_{L^2}\varphi ^k\rightarrow u_0^{\delta }\) and \(\theta ^m(0)=\sum _{k=1}^m(\theta _0^{\delta },\phi ^k)_{L^2}\phi ^k\rightarrow \theta _0^{\delta }\) in \(H^2\). Combining (3.8), (3.9), (3.23) and (3.34) together, we derive that
and
Hence, there exists a sequence \(\{m_k\}\) such that \((\nabla u^{m_k},u_t^{m_k},\nabla \theta ^{m_k},\theta _t^{m_k})\rightharpoonup ^*(\nabla u^{\delta },u_t^{\delta },\nabla \theta ^{\delta },\theta _t^{\delta })\) in \(L^{\infty }(0,T^*;L^2)\) and \((u_t^{m_k},\theta ^{m_k}_t) \rightharpoonup (u_t^{\delta },\theta _t^{\delta })\) in \(L^2(0,T^*;H^1)\) for some limit \((u^{\delta },\theta ^{\delta })\). In view of (3.8), (3.9), (3.23) and (3.34), we derive that
and
It is easy to find a sequence \(\delta ^k\in (0,1)\), which satisfies \(\delta ^k\rightarrow 0\), such that \((u^{\delta _k},\theta ^{\delta _k}) \rightharpoonup ^*(u,\theta )\) in \(L^{\infty }(0,T^*;H^1)\) and \((u_t^{\delta _k},\theta _t^{\delta _k})\rightharpoonup (u_t,\theta _t)\) in \(L^2(0,T^*;H^1)\) for some \((u,\theta )\) satisfying
and
Now it is a simple matter to show that \((u,\theta )\) is a weak solution to the original problem with the initial data \((u_0,\theta _0)\).
For the higher regularity of \((u,\theta ,p)\), by (3.10) and (3.21), we have
Applying Lemma 2.1, it yields that
Hence, \(\nabla u,\nabla \theta , p\in L^{\infty }(0,T;H^1)\bigcap L^2(0,T;H^2)\) and \(u\in L^{\infty }(0,T;W^{2,r})\). The time continuity of \((u,\theta )\) in \(H^2\) follows from a standard embedding result. This completes the proof. \(\square \)
4 Proof of Theorem 1.1
In this section, we first prove the existence of local strong solution to problem (1.1)–(1.4) for the case of a positive initial density and obtain the uniform bounds which are independent of the lower bounds of the initial density. Then, by applying standard regularizing techniques and compactness arguments, these uniform bounds will be used to prove the existence of strong solutions for the general cases.
The result on the existence of local strong solution to problem (1.1)–(1.4) for the case of a positive initial density will be given in the following Proposition 4.1:
Proposition 4.1
Let \(q\in (3,\infty )\) be a fixed constant and the initial data \((\rho _0,u_0,\theta _0)\) satisfy the regularity conditions
and the compatibility conditions
for some \(p_0\in H^1\) and \(g_1,~g_2\in L^2\). Then, there exist a positive time \(T>0\) and a unique strong solution \((\rho ,u,\theta ,p)\) for problem (1.1)–(1.4) such that
for some \(r\in (3,\min \{6,q\})\).
In order to prove Proposition 4.1, we construct approximate solutions, inductively, as follows:
-
(1).
define \(u^0=0\), \(d^0=d_0\) and \(\theta ^0=0\);
-
(2).
assuming that \(u^{k-1}\), \(d^{k-1}\) and \(\theta ^{k-1}\) was defined for \(k\ge 1\), let \((\rho ^k,u^k,d^k,\theta ^k,p^k)\) be the unique solution to the following initial boundary value problem
$$\begin{aligned} \left\{ \begin{aligned}&\rho ^k_t+u^{k-1}\cdot \nabla \rho ^k=0, \quad \hbox {in}~\Omega \times (0,T),\\&\rho ^k u^{k}_t+\rho ^k u^{k-1}\cdot \nabla u^k-\nabla \cdot (2\mu (\rho ^k)D(u^k)) +\nabla p^k=\nabla \cdot (\nabla d^{k }\odot \nabla d^{k }), \\&d_t^k-\Delta d^k=|\nabla d^{k-1}|^2d^{k-1}- u^{k-1}\cdot \nabla d^{k-1},\\&c_{\nu }[\rho ^k\theta ^k_t+\rho u^{k-1}\cdot \nabla \theta ^k]-\nabla \cdot (\kappa (\rho ^k)\nabla \theta ^k)\\&\qquad =2\mu (\rho ^k)|D(u^{k-1})|^2+\rho ^k|\Delta d^{k-1} +|\nabla d^{k-1}|^2d^{k-1}|^2,\\&\nabla \cdot u^k=0, \end{aligned} \right. \end{aligned}$$(4.2)
Then we show that the approximate solutions satisfy some uniform bounds and converge to a local strong solution to the original nonlinear problem.
4.1 Uniform Bounds
In order to prove the uniform bounds of the approximate solutions, we define
for \(K\ge 1\). We will estimate each term of \(\Phi _K(t)\) in terms of some integrals of \(\Phi _K(t)\) and thus prove that \(\Phi _K(t)\) is locally bounded. Observe that
Lemma 4.2
([23]) There exists a positive constant \(N=N(q)\) such that
for all \(k\in [1,K]\).
Lemma 4.3
There exists a positive constant \(N=N(q)\) such that
for any \(k\in [1,K]\).
Proof
It follows from Poincaré’s inequality and (4.3) that
Integrating (4.2)\(_4\) over (0, t), we arrive at
For any fixed \(\tau \in (0,t)\), integrating (4.6) over \((\tau ,t)\subset [0,T]\), it yields that
Letting \(\tau \rightarrow 0^+\), we deduce that
Denote \(\overline{\theta ^k}=\frac{1}{|\Omega |}\int \theta ^kdx\). Applying Poincaré’s inequality, we obtain
Hence,
Similarly, we also have
Multiplying (4.2)\(_4\) by \(\theta _t^k\), integrating over \(\Omega \), it yields that
Note that the Taylor’s expansion of \(e^x\) is
By using (4.3), (4.5) and the Taylor’s expansion of \(e^x\), we deduce that
Moreover, we also have
and
Adding (4.11)–(4.15) together gives
Note that
where we have used the Taylor’s expansion of \(e^x\). Owning to (4.16)-(4.17) and Taylor’s expansion of \(e^x\), we derive that
This completes the proof. \(\square \)
Lemma 4.4
There exists a positive constant \(N=N(q)\) such that
for any \(k\in [1,K]\).
Proof
We remark that (4.5) can be rewrite as
Note that
which implies
where we have used the Taylor’s expansion of \(e^x\). Similarly, we also have
In order to obtain the estimate (4.4), we also need to estimate \(\Vert d_t^{k-1}\Vert _{L^2}\) and \(\Vert \nabla d_t^{k-1}\Vert _{L^2}\). Note that
we have
and
Differentiating (4.2)\(_3\) with respect to t, multiplying by \(\theta _t^k\), integrating over \(\Omega \), it yields that
Applying Lemma 4.3, (4.3), (4.5), (4.20), (4.21) and (4.22), we estimate \(\tilde{J}_1\)–\(\tilde{J}_8\) one by one.
and
Combining (4.25)–(4.33) together gives
Applying Taylor’s expansion for \(e^x\), integrating over (0, t) yields
This completes the proof. \(\square \)
On the basis of Lemmas 4.2–4.4, we have
for some \(N=N(q)>0\). Hence, thanks to this integral inequality, we show that there is a time \(T_0\in (0,T)\), depending only on the parameter of C (independent of K) such that
In addition, Lemmas 4.2–4.4 yield the following uniform bounds:
for all \(k\ge 1\).
4.2 Convergence
We will show that the whole sequences \((\rho ^k,u^k,d^k,\theta ^k)\) of approximate solutions converge to a strong solution of problem (1.1)–(1.4) in a strong sense. Define
Firstly, we consider the transport equation (4.2)\(_1\). Note that
Multiplying by \(\varrho ^{k+1}\), integrating over \(\Omega \) yields (see [23])
Secondly, we consider the linearized momentum equations (4.2)\(_2\), which satisfies
Multiplying by \(\vartheta ^{k+1}\), integrating over \(\Omega \) yields (see [23])
Thirdly, the linearized equation (4.2)\(_3\) satisfies
Multiply by \(\Delta \chi ^{k+1}\) and integrate by parts over \(\Omega \). This yields that (see [23])
Fourthly, we consider the linearized equation (4.2)\(_4\), which satisfies
Multiplying by \(\upsilon ^{k+1}\), integrating over \(\Omega \) yields
We estimate \(\tilde{K}_1\)–\(\tilde{K}_5\) one by one. Applying (2.3), we obtain
and
Owning to (4.35)–(4.44), we arrive at
Now, for a small fixed \(\varepsilon >0\), let us define
We obtain
In view of Gronwall’s inequality, we deduce that
Choosing \(\delta >0\) sufficiently small, using (4.46), we deduce that [6]
On the other hand, a simple calculation shows that
which obviously implies that
as \(k\rightarrow \infty \) for some limits \(\rho \), u, d and \(\theta \).
Now, it is a simple matter to prove \((\rho ,u,d,\theta )\) is a weak solution to problem (1.1)–(1.4) with positive initial density. Using the lower semi-continuity of norms, we know that \((\rho ,u,d,\theta )\) satisfies
4.3 Proof of Theorem 1.1
Suppose that \((\rho _0,u_0,d_0,\theta _0)\) satisfies the assumption in Theorem 1.1. Therefore, for each \(\delta >0\), let \(\rho _0^{\delta }=\rho _0+\delta \) satisfies
Let \((u^{\delta },\theta ^{\delta })\in H_0^1\times H^1\) be a solution to
By virtue of the regularity estimate, we arrive at
Hence, on the basis of Proposition 4.1, there is a time \(T_0\in (0,T)\) and a unique strong solution \((\rho ^{\delta } ,u^{\delta },d^{\delta },\theta ^{\delta },p^{\delta })\) in \([0,T_0]\times \Omega \) to the problem with initial data replaced by \((\rho _0^{\delta },u_0^{\delta },d^{\delta }_0,\theta _0^{\delta })\). Since \((\rho ^{\delta },u^{\delta },d^{\delta },\theta ^{\delta }, p^{\delta })\) satisfies the regularity estimate with C independent of \(\delta \). Let \(\delta \rightarrow 0^+\), it is easy to see that \((\rho ,u,d,\theta ,p)\) is a strong solution, which satisfies
This completes the proof.
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Zhao, X., Zhu, M. Strong Solutions to the Density-Dependent Incompressible Nematic Liquid Crystal Flows with Heat Effect . Bull. Malays. Math. Sci. Soc. 44, 1579–1611 (2021). https://doi.org/10.1007/s40840-020-01026-2
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DOI: https://doi.org/10.1007/s40840-020-01026-2