Abstract
In this paper, we consider an initial and boundary value problem to the three-dimensional (3D) nonhomogeneous nematic liquid crystal flows with density-dependent viscosity and vacuum. Combining delicate energy method with the structure of the system under consideration, the global well-posedness of strong solutions is established, provided that \(\Vert \rho _{0}\Vert _{L^{1}}+\Vert \nabla \varvec{d}_0\Vert _{L^2}\) is suitably small. In particular, the initial velocity can be arbitrarily large. Moreover, the exponential decay rates of the strong solution are also obtained.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and main result
Let \(\Omega \subset \mathbb {R}^3\) be a bounded smooth domain. The motion of the nonhomogeneous nematic liquid crystal flows is governed by the following simplified version of the Ericksen–Leslie equations with density-dependent viscosity in \(\Omega \times (0, T]\):
where \(\rho , \varvec{u}, \varvec{d}\) and P are the density of the fluid, velocity, macroscopic average of the nematic liquid crystal orientation and pressure, respectively. The deformation tensor \(\mathfrak {D}(\varvec{u})\) is given by
The viscosity coefficient \(\mu =\mu ({\rho })\) is a general function of density, which is assumed to satisfy
for some positive constant \(\underline{\mu }.\) The notation \(\nabla \varvec{d} \odot \nabla \varvec{d}\) denotes the \(3\times 3\) matrix whose ij component is given by \(\partial _i \varvec{d} \cdot \partial _j \varvec{d}, i,j=1,2,3.\)
We seek for the solutions to the system (1.1) with the following initial and boundary conditions:
Here \(\varvec{d}_0: \overline{\Omega } \rightarrow \textbf{S}^2\) is a given vector satisfying \(\nabla \varvec{d}_0 = \varvec{0}\) on the boundary \(\partial \Omega \) (see more details of this fact in [9]).
The above system (1.1) describes the macroscopic evolution for the nematic liquid crystals. It is a simplified version of the Ericksen–Leslie model [4, 11], but it still retains most important mathematical structures as well as most of the essential difficulties of the original Ericksen–Leslie model. For more details on the hydrodynamic continuum theory of liquid crystals, we refer the readers to the monographs [1, 6]. Mathematically, system (1.1) is a coupling between the nonhomogeneous incompressible Navier–Stokes equations and the transported heat flows of harmonic map, and thus, its mathematical analysis is full of challenges.
When \(\varvec{d}\) is a constant vector satisfying \(|\varvec{d}|=1\), system (1.1) reduces to the nonhomogeneous incompressible Navier–Stokes equations with density-dependent viscosity. As pointed out in many papers, the strong interaction between density and velocity will bring some difficulties in the mathematical analysis due to the density-dependent viscosity. When the initial vacuum is taken into account, Lions [17] established the global existence of weak solutions to the nonhomogeneous incompressible Navier–Stokes equations. Later, Cho and Kim [2] constructed a unique local strong solution by imposing the following compatibility condition on the initial data:
for some \((P_0, \varvec{g}) \in H^1 \times L^2.\) Recently, Huang and Wang [10], and independently by Zhang [29], obtained the global existence and uniqueness of strong solution of Navier–Stokes equations provided that \(\Vert \nabla \varvec{u}_0\Vert _{L^2}\) is suitably small.
Let us come back to the system (1.1). Compared with the nonhomogeneous incompressible Navier–Stokes equations, due to the strong coupling and interaction between the fluid motion and the macroscopic orientation vector, the mathematical analysis on the system (1.1) will become more subtle. When the viscosity \(\mu \) is a positive constant, there is a huge literature on the studies of the well-posedness of solutions to (1.1). For the initial density away from vacuum, Wen and Ding [26] established the global existence and uniqueness of strong solution to the 2D problem with small initial energy \(\Vert \sqrt{\rho _0}\varvec{u}_0\Vert _{L^2}^2 + \Vert \nabla \varvec{d}_0\Vert _{L^2}^2\). J. Li [12] obtained the same result of the 2D problem for large initial data under a geometric condition the initial direction field \(\varvec{d}_0=(d_{01}, d_{02}, d_{03})\):
Meanwhile, X. Li and Wang [16] obtained the global strong solution for small initial data, and they also established the weak–strong uniqueness. On the other hand, if the initial density allows to vanish, Wen and Ding [26] established the local well-posedness of strong solution under the assumption that the initial data satisfy a similar compatibility condition as (1.5). Ding et. al [3] and J. Li [13] extended this local strong solution in 3D to global in time for some small initial data. Yu and Zhang [28] established the global well-posedness of strong solution in 3D for small initial energy with Nuemann boundary condition for the macroscopic orientation field. Recently, assuming that the initial orientation field satisfies a geometric condition (1.6), Liu and Zhang [19] and Liu et. al [18] established the global well-posedness of strong solution to the 2D Cauchy problem with large initial data for positive and zero far field density at space infinity, respectively. Meanwhile, Li et. al [14] obtained the same result under small initial data for the 2D Cauchy problem with zero far field density if the initial density decays not too slow at infinity. When the viscosity is a function of density, the analysis becomes more difficult due to the strong coupling of viscosity and velocity field. Gao et. al [7] established the local well-posedness of strong solution in a bounded domain under a compatibility condition on the initial data. In addition, they also obtained the Serrin-type blow-up criterion. Subsequently, Liu [20, 21] proved the global existence and uniqueness of strong solution in 2D/3D for some small initial data (see also Liu and Zhong [22]). Very recently, Ye and Zhu [27] established the global well-posedness of strong solution under the initial norm \(\Vert \varvec{u}_0\Vert _{\dot{H}^{\alpha }} + \Vert \nabla \varvec{d}_0\Vert _{\dot{H}^{\alpha }}(1/2<\alpha \le 1)\) being suitably small.
The purpose of this paper is to establish the global strong solutions to the 3D incompressible nematic liquid crystal flows with density-dependent viscosity, provided that \(\Vert \rho _0\Vert _{L^1} + \Vert \nabla \varvec{d}_0\Vert _{L^2}\) is suitably small allowing large oscillation of the velocity.
Before stating our main result, we first explain the notations and conventions used throughout this paper. For \(p \in [1, \infty ]\) and integer \(k \in \mathbb {N}_+\), we use \(L^p = L^p(\Omega )\) and \(W^{k,p}= W^{k,p}(\Omega )\) to denote the standard Lebesgue and Sobolev spaces, respectively. When \(p=2,\) we use \(H^k=W^{k,2}(\Omega )\). The space \(H^{1}_{0,\sigma }\) stands for the closure in \(H^1\) of the space \(C^{\infty }_{0,\sigma } \triangleq \{\varvec{\phi } \in C^{\infty }_0 | \textrm{div} \varvec{\phi } =0\}.\) And for two \(3 \times 3\) matrices \(A=\left( A_{ij}\right) \) and \(B=\left( B_{ij}\right) \), we denote by
Now we state our main result for the problem (1.1)-(1.4) as follows.
Theorem 1.1
For \(\bar{\rho }>0\) and \(q \in (3,\infty )\), assume that the initial data \((\rho _{0}, \varvec{u}_{0}, \varvec{d}_{0})\) satisfies
Then, there exists a small positive constant \(\varepsilon _{0}\) depending only on \(\Omega , \overline{\mu }\triangleq \sup \limits _{[0, \bar{\rho }]}\mu (\rho ), \underline{\mu },\) \(q, \bar{\rho }, \Vert \nabla \mu (\rho _0)\Vert _{L^q}, \Vert \nabla \varvec{u}_0\Vert _{L^2}\) and \(\Vert \nabla ^2 \varvec{d}_0\Vert _{L^2}\), such that if
the problem (1.1)-(1.4) admits a unique global strong solution \((\rho ,\varvec{u}, \varvec{d}, P)\) satisfying, for any \(3< r < \min \{6,q\}\) and \(\tau >0\),
Moreover, it holds that
and there exists some positive constant C depending only on \(\Omega , \overline{\mu }, \underline{\mu }, q, \bar{\rho },\) \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}, \Vert \nabla \varvec{u}_0\Vert _{L^2}\) and \(\Vert \nabla ^2 \varvec{d}_0\Vert _{L^2}\), such that, for \(t \ge 1,\)
Here, \(\sigma \triangleq \min \{\frac{\underline{\mu }}{\bar{\rho }l^2}, \frac{1}{l^2}\}\) with l being the diameter of \(\Omega .\)
Remark 1.1
Since \(\Omega \) is a bounded smooth domain, we deduce from Hölder’s inequality that
Thus, our Theorem 1.1 improves Liu’s result [20]. Moreover, by modifying the proof of this paper slightly, similar result holds true for the case of 2D bounded domains. Hence, we also generalize Liu’s result [21].
We mainly use the continuation argument to give a proof of Theorem 1.1. Since the local strong solution was obtained by Lemma 2.1, the key issue is to establish global a priori estimates on strong solutions to (1.1)-(1.4) in suitable higher-order norms. Due to the strong interaction between viscosity and velocity, the method used for the constant viscosity case cannot be applied here directly. Moreover, compared with the previous work on nonhomogeneous Navier–Stokes equations with density-dependent viscosity [28, 31], the proof of Theorem 1.1 is much more involved due to the strong coupling between the velocity and the macroscopic orientation vector. Hence, some new ideas are needed to overcome these difficulties.
Firstly, motivated by the work of [8], we find that \(\Vert \sqrt{\rho }\varvec{u}\Vert _{L^2}^2\) and \(\Vert \nabla \varvec{d}\Vert _{L^2}^2\) decay at the rate of \(e^{\sigma t}\) for some \(\sigma \) depending only on \(\bar{\rho }, \underline{\mu }\) and \(\Omega \) with the help of Poincaré’s inequality and Sobolev’s inequality (see (3.8)). Next, we attempt to obtain the uniform in time-weighted estimates of \(\Vert \nabla \varvec{u}\Vert _{L^2}^2\) and \(\Vert \nabla ^2 \varvec{d}\Vert _{L^2}^2\). To overcome the difficulties caused by the density-dependent viscosity and strongly coupling between the velocity and macroscopic orientation vector, we assume the condition (3.2) holds. Moreover, regularity properties of the Stokes system and elliptic equations play important roles. Then we obtain the desired bounds of \(\Vert \nabla \varvec{u}\Vert _{L^2}^2\) and \(\Vert \nabla ^2 \varvec{d}\Vert _{L^2}^2\)(see Lemma 3.3). These bounds are crucial in deriving time-weighted estimates of \(L^{\infty }(0, T; L^2)\)-norms of \(\sqrt{\rho }\varvec{u}_t\) and \(\nabla \varvec{d}_t\). The next step is to show the quantity \(\Vert \nabla \mu (\rho )\Vert _{L^q}\) is in fact less than \(2\Vert \nabla \mu (\rho _0)\Vert _{L^q}\). To this end, it needs to deal with \(\Vert \nabla \varvec{u}\Vert _{L^1(0, T; L^{\infty })}\). Based on the time-weighted estimates (Lemmas 3.1-3.4), we find that the uniform bound (with respect to time) on the \(L^1(0, T; L^{\infty })\)-norm of \(\nabla \varvec{u}\) is bounded by the initial mass and \(L^2\)-norm of \(\nabla \varvec{d}_0\). This completes the proof of (3.3) provided that the assumption (1.8) stated in Theorem 1.1 holds. Finally, the higher-order estimates on solutions are obtained (see Lemmas 3.6-3.7).
The remaining parts of this paper are arranged as follows. In Section 2, we shall give some auxiliary lemmas which are useful in later analysis. In Section 3, we establish some necessary a priori estimates to extend the local strong solution. Finally, we give the proof of the main result Theorem 1.1 in Section 4.
2 Preliminaries
In this section, we shall recall some known facts and elementary inequalities that will be used extensively later.
We start with the local existence and uniqueness of strong solutions whose proof can be performed in a similar way as [15, 24].
Lemma 2.1
Assume that \((\rho _{0},\varvec{u}_{0},\varvec{d}_{0})\) satisfies (1.7). Then, there exist a small time \(T_{0}>0\) and a unique strong solution \((\rho , \varvec{u}, \varvec{d}, P)\) to the problem (1.1)-(1.4) in \(\Omega \times (0,T_{0}]\).
Next, the following Gagliardo–Nirenberg inequality (see [23, Theorem 10.1, p.27]) will be useful in the next section.
Lemma 2.2
For \(p \in [2, 6], q \in (1, +\infty ),\) and \(r \in (3, +\infty ),\) there exists some generic constant C which may depend only on p, q and r, such that for \(f \in H^1_0, g \in L^q \cap D^{1, r},\) the following inequalities hold.
and
Finally, the following regularity results for the Stokes system will be used frequently in deriving the higher-order estimates. Refer to [10, Lemma 2.1] for the proof.
Lemma 2.3
For constants \(q>3, \underline{\mu }, \bar{\mu } >0,\) in addition to (1.2), the function \(\mu \) satisfies
Assume that \((\varvec{u}, P) \in H^1_{0,\sigma } \times L^2\) is the unique weak solution to the following problem
Then, there exists a positive constant C depending only on \(\Omega , \underline{\mu }\) and \(\bar{\mu }\) such that the following regularity results hold true:
-
If \(\textbf{F} \in L^2,\) then \((\varvec{u}, P) \in H^2 \times H^1\) and
$$\begin{aligned} \Vert \varvec{u}\Vert _{H^2} + \Vert P/\mu (\rho )\Vert _{H^1} \le C \Vert \textbf{F}\Vert _{L^2}\left( 1 + \Vert \nabla \mu (\rho )\Vert _{L^q}^{\frac{q}{q-3}}\right) . \end{aligned}$$(2.4) -
If \(\textbf{F} \in L^r\) for some \(r \in (2, q)\), then \((\varvec{u}, P) \in W^{2,r} \times W^{1,r}\) and
$$\begin{aligned} \Vert \varvec{u}\Vert _{W^{2,r}} + \Vert P/\mu (\rho )\Vert _{W^{1,r}} \le C \Vert \textbf{F}\Vert _{L^r}\left( 1 + \Vert \nabla \mu (\rho )\Vert _{L^q}^{\frac{q(5r-6)}{2r(q-3)}}\right) . \end{aligned}$$(2.5)
3 A priori estimates
In this section, we will establish some necessary a priori bounds for strong solution \((\rho , \varvec{u}, \varvec{d})\) of the problem (1.1)-(1.4) to extend the local strong solution guaranteed by Lemma 2.1. Thus, let \(T>0\) be a fixed time and \((\rho ,\varvec{u},\varvec{d})\) be the strong solution to (1.1)-(1.4) on \(\Omega \times (0,T]\) with initial data \((\rho _{0},\varvec{u}_{0},\varvec{d}_{0})\) satisfying (1.7). Before proceeding further, we rewrite another equivalent form of the system (1.1) as follows.
In what follows, we denote by
We give the following key a priori estimates on \((\rho ,\varvec{u},\varvec{d},P)\).
Proposition 3.1
There exists some positive constant \(\varepsilon _{0}\) depending only on \(\Omega , \overline{\mu }, \underline{\mu }, q, \bar{\rho },\) \(\Vert \nabla \mu (\rho _0)\Vert _{L^q}, \Vert \nabla \varvec{u}_0\Vert _{L^2}\) and \(\Vert \nabla ^2 \varvec{d}_0\Vert _{L^2}\) such that if \((\rho , \varvec{u}, \varvec{d})\) is a strong solution to (1.1)-(1.4) on \(\overline{\Omega } \times (0,T]\) satisfying
then the following estimates hold
provided that
Here, \(m_0 \triangleq \Vert \rho _0\Vert _{L^1}\) denotes the initial total mass.
The proof of Proposition 3.1 consists of a series of lemmas. In the following, we will use the convention that C denotes some generic positive constant which may depend on \(\Omega , \overline{\mu }, \underline{\mu }, q, \bar{\rho }\) and initial data.
First of all, due to the transport equation (3.1)\(_1\), we have the following estimate on the \(L^{\infty }(0, T; L^{\infty })\)-norm of the density, whose proof can be found in [17, Theorem 2.1].
Lemma 3.1
It holds that for any \(t \in [0, T],\)
Next, we give the following standard energy estimate of the system (3.1), which reads as follows.
Lemma 3.2
Under the condition (3.2), it holds that
provided that
where \(C_1\) is defined as in (3.12) depending only on \(\Omega \). Moreover, for \(\sigma \triangleq \min \left\{ \frac{\underline{\mu }}{\bar{\rho }l^2}, \frac{1}{l^2}\right\} \) with l being the diameter of \(\Omega \), one has that
Proof
Multiplying (3.1)\(_{2}\) by \(\varvec{u}\) and integrating over \(\Omega \), we deduce from integration by parts that
Multiplying (3.1)\(_{3}\) by \(-(\Delta \varvec{d}+|\nabla \varvec{d}|^{2}\varvec{d})\) and integrating over \(\Omega \) yield
Combining (3.10) and (3.9), we obtain that
Noting that
and
we thus deduce from (3.11), (1.2), (3.2) and the Gagliardo–Nirenberg inequality that
for some constant \(C_1\) depending only on \(\Omega \). Thus, we obtain that
provided that
Integrating (3.13) over [0, T] implies
due to the Gagliardo–Nirenberg inequality and (1.7).
It follows from Poincaré’s inequality (see [25, (A.3), p.266] that
where l is the diameter of \(\Omega \). Hence, we get
Consequently, letting \(\sigma \triangleq \min \left\{ \frac{\underline{\mu }}{2\overline{\rho }l^{2}},\frac{1}{2\,l^{2}}\right\} \), then we derive from (3.13) and (3.17) that
Multiplying (3.18) by \(e^{\sigma t}\), one has
Thus, integrating (3.19) with respect to t gives (3.8), which combined with (3.15) completes the proof of Lemma 3.2.\(\square \)
Next, we will derive important (time-weighted) estimates on the spatial gradients of the strong solution \((\varvec{u}, \nabla \varvec{d}).\)
Lemma 3.3
Under the conditions (3.2) and (3.7), if
where \(C_2\) is defined as in (3.35) depending only on \(\Omega \) and \(\underline{\mu }\), then
Furthermore, for \(i=\{1, 2\}\) and \(\sigma \) as in Lemma 3.2, one has that
Proof
1. Since \(\mu (\rho )\) is a continuously differentiable function, we deduce from (3.1)\(_{1}\) that
Multiplying (3.1)\(_{2}\) by \(\varvec{u}_{t}\), and integrating by parts over \(\Omega \), we have
First, we obtain from (3.24) that
Then, inserting (3.26) into (3.25), it follows from integration by parts that
Now, we are ready to estimate terms \(I_1\)-\(I_3\). By Hölder’s inequality and the Gagliardo–Nirenberg inequality, we get
By Hölder’s inequality, the Gagliardo–Nirenberg inequality and (3.5), we obtain
By Sobolev’s inequality and (3.2), we have
Substituting (3.28)-(3.30) into (3.27) leads to
2. Multiplying (3.1)\(_3\) by \(\Delta \varvec{d}_t\) and integrating the resulting equation over \(\Omega \), it follows from the Gagliardo–Nirenberg inequality, Hölder’s inequality and (1.4) that
which together with (3.31) gives rise to
where
By Hölder’s inequality and Sobolev’s inequality, we have
for some positive constant \(C_2\) depending only on \(\Omega \) and \(\underline{\mu }\). Thus, we obtain that
provided that
3. Recall that \((\varvec{u},P)\) satisfies the following density-dependent Stokes system:
Applying Lemma 2.3 with \(\textbf{F}=-\rho \varvec{u}_{t}-\rho \varvec{u}\cdot \nabla \varvec{u}-\textrm{div}(\nabla \varvec{d}\odot \nabla \varvec{d})\), we obtain from (3.2), (3.5) and the Gagliardo–Nirenberg inequality that
which implies that
Taking \(\nabla \) operator to the equation (3.1)\(_3\), one has
It follows from \(L^2\) estimates of the elliptic system (3.41), it is easy to deduce from (3.1)\(_3\) that
which gives
This along with (3.40) leads to
Inserting (3.43) and (3.44) into (3.33), and applying Young’s inequality, we deduce that
Applying Gronwall’s inequality, (3.2), (3.6) and (3.36) leads to
For \(i=\{1, 2, 3\},\) multiplying (3.45) by \(t^i,\) we obtain from (3.46) and (3.36) that
For \(\sigma \) as in Lemma 3.2, and any \(k \in \mathbb {N}\), we derive from (3.8) that
Integrating (3.47) over [0, T] together with (3.48) leads to (3.22). Moreover, multiplying (3.45) by \(e^{\sigma t}\), we deduce from (3.46) and (3.36) that
Integrating the above inequality over [0, T] together with (3.8) and (3.36) gives (3.23). Therefore, the proof of lemma 3.3 is completed.\(\square \)
Remark 3.1
Combining (3.43) and (3.44), we have
And it follows from (3.1)\(_3\), (3.21) and the Gagliardo–Nirenberg inequality that
which together with (3.21), (3.6) and (3.50) implies, for \(i=\{1, 2\}\)
Lemma 3.4
Under the conditions (3.2), (3.7) and (3.20), for \(i \in \{1, 2\},\) it holds that
Moreover, for \(\sigma \) as in Lemma 3.2 and \(\zeta (t) \triangleq \min \{1, t\},\) one has that
Proof
1. Differentiating (3.1)\(_{2}\) with respect to time variable t gives
Multiplying (3.55) by \(\varvec{u}_{t}\) and integrating the resulting equality by parts over \(\Omega \), we deduce from (3.1)\(_1\) that
By Hölder’s inequality, Sobolev’s inequality, the Gagliardo–Nirenberg inequality, (3.5), (3.21) and (3.24), we obtain that
Substituting the above estimates of \(J_{1}\)-\(J_5\) into (3.56) and noting that
we obtain from (3.50), (3.21) and Young’s inequality that
2. Differentiating (3.1)\(_3\) with respect to t, and multiplying the resulting equality by \(\varvec{d}_t\), we obtain from integration by parts over \(\Omega \) that
By Hölder’s inequality, Sobolev’s inequality, the Gagliardo–Nirenberg inequality, (3.5) and (3.21), we obtain that
Hence,
Differentiating (3.41) with respect to t gives
Multiplying (3.61) by \(\nabla \varvec{d}_{t}\), we obtain from integration by parts that
By Hölder’s inequality, Sobolev’s inequality and the Gagliardo–Nirenberg inequality, we obtain from (3.5) and (3.21) that
Substituting the above estimates of \(K_1\)-\(K_5\) into (3.62), we arrive at
Adding the resulting inequality with (3.58) and (3.60), and choosing \(\delta \) suitably small, we deduce that
Multiplying (3.64) by \(t^i(i \in \{1,2,3\})\) gives that
which together with Gronwall’s inequality, (3.48), (3.21), and (3.22) leads to (3.53). Furthermore, multiplying (3.64) by \(e^{\sigma t}\) gives that
which combined with Gronwall’s inequality (3.21), (3.8) and (3.23) implies (3.54).\(\square \)
Lemma 3.5
Under the conditions (3.2), (3.7) and (3.20), there exists a positive constant C depending only on \(\Omega , q, \bar{\rho }, \underline{\mu }, \bar{\mu }, \Vert \nabla \mu (\rho _0)\Vert _{L^q}, \Vert \nabla \varvec{u}_0\Vert _{L^2}\) and \(\Vert \nabla ^2 \varvec{d}_0\Vert _{L^2}\) such that
Proof
For \(3< r < \min \left\{ q, 6\right\} \), we deduce from Lemma 2.3, (3.2), (3.5), Hölder’s inequality, Sobolev’s inequality and the Gagliardo–Nirenberg inequality that
which combined with Sobolev’s inequality gives that
If \(0< T\le 1\), we derive from (3.53) and Hölder’s inequality that
If \(T>1\), due to \(3< r < \min \{6,q\},\) we deduce from (3.53) and Hölder’s inequality that
which along with (3.70) yields that, for any \(T>0\),
It follows from (3.40), (3.2), (3.21) and (3.6) that
and
With the estimates (3.72)-(3.74), integrating (3.69) on [0, T] gives
This completes the proof of Lemma 3.5.\(\square \)
Proof of Proposition 3.1
Taking the spatial gradient operator \(\nabla \) on the transport equation (3.24) implies
Multiplying (3.76) by \(q|\nabla \mu ({\rho })|^{q-2}\nabla \mu ({\rho })\) and integrating the resulting equation over \(\Omega \) give
which combined with Gronwall’s inequality and (3.67) leads to
for some constant \(C_3\) depending only on \(\Omega , q, \bar{\rho }, \underline{\mu }, \bar{\mu }, \Vert \nabla \mu (\rho _0)\Vert _{L^q}, \Vert \nabla \varvec{u}_0\Vert _{L^2}\) and \(\Vert \nabla ^2 \varvec{d}_0\Vert _{L^2}\). This implies that
provided \(m_{0}\le \varepsilon _{1} \triangleq \min \left\{ 1, \left( \frac{1}{2C_1}\right) ^9, \left( \frac{1}{8C_2}\right) ^9, \left( \frac{\log 2}{2C_{3}}\right) ^{6}\right\} \).
Next, it follows from (3.6) and (3.21) that
for some constant \(C_4\) depending only on \(\Omega , q, \bar{\rho }, \underline{\mu }, \bar{\mu }, \Vert \nabla \mu (\rho _0)\Vert _{L^q}, \Vert \nabla \varvec{u}_0\Vert _{L^2}\) and \(\Vert \nabla ^2 \varvec{d}_0\Vert _{L^2}\). This yields that
provided \(m_{0}\le \varepsilon _{2} \triangleq \min \left\{ \left( \frac{1}{2C_1}\right) ^9, \left( \frac{1}{8C_2}\right) ^9, (\frac{1}{C_{4}})^{3}\right\} \).
Finally, multiplying (3.41) by \(4|\nabla \varvec{d}|^{2}\nabla \varvec{d}\) and integrating by parts over \(\Omega \) give rise to
which implies
This along with Gronwall’s inequality and (3.2) yields that
Subsequently, it follows from (3.6), (3.84) and Hölder’s inequality that
for some constant \(C_5\) depending only on \(\Omega , q, \bar{\rho }, \underline{\mu }, \bar{\mu }, \Vert \nabla \mu (\rho _0)\Vert _{L^q}, \Vert \nabla \varvec{u}_0\Vert _{L^2}\) and \(\Vert \nabla ^2 \varvec{d}_0\Vert _{L^2}\). This yields that
provided \(m_{0} \le \varepsilon _{3} \triangleq \min \left\{ \left( \frac{1}{2C_1}\right) ^9, \left( \frac{1}{8C_2}\right) ^9, (\frac{1}{C_{5}})^{6}\right\} \).
As a consequence, if
we derive (3.2) from (3.79), (3.75) and (3.85). Therefore, the proof of Proposition 3.1 is complete. \(\square \)
Lemma 3.6
Under the conditions (3.2) and (3.20), there exists a positive C depending only on \(\Omega , q,\overline{\mu },\underline{\mu },\bar{\rho },\Vert \nabla \varvec{u}_{0}\Vert _{L^{2}}, \Vert \nabla ^{2}\varvec{d}_{0}\Vert _{L^{2}}\) and \(\Vert \nabla \mu ({\rho _0})\Vert _{L^{q}}\) such that
Proof
By an argument similar to the one used in (3.79), we obtain from (1.7) that
It follows from (3.1)\(_1\), Hölder’s inequality and Sobolev’s inequality that
which together with (3.89) and (3.21) yields that
This completes the proof of Lemma 3.6.\(\square \)
Lemma 3.7
Under the conditions (3.2),(3.7) and (3.20), there exists a positive constant C depending on \(\Omega , q,\overline{\mu },\underline{\mu },\bar{\rho },\Vert \nabla \varvec{u}_{0}\Vert _{L^{2}}, \Vert \nabla ^{2}\varvec{d}_{0}\Vert _{L^{2}}\) and \(\Vert \nabla \mu ({\rho _0})\Vert _{L^{q}} \) such that for \(r \in (3, \min \{q,6\})\),
Furthermore, for \(\sigma \) as in Lemma 3.2 and \(\zeta (t)\) as in Lemma 3.4, one has that
Proof
We obtain from (3.50) and (3.21) that
which together with (3.22) and (3.53) implies
And it follows from (3.93), (3.53) and (3.23) that
For \(3<r<\min \left\{ q,6\right\} \), we get from (3.68) and (3.93) that
which combined with (3.6), (3.21), (3.22) and (3.53) leads to
Finally, it follows from \(L^2\)-theory of elliptic equations, (3.21) and Sobolev’s inequality that
which along with (3.22), (3.53), (3.48), (3.94) and (3.52) implies that
Therefore, the proof of Lemma 3.7 is complete.\(\square \)
4 proof of Theorem 1.1
With all the a priori estimates obtained in Sect. 2 at hand, we are now in a position to give a proof of Theorem 1.1.
Proof of Theorem 1.1
First, by Lemma 2.1, there exists a \(T_*>0\) such that the initial and boundary value problem (1.1)-(1.4) admits a unique local strong solution \((\rho , \varvec{u}, \varvec{d}, P)\) on \(\Omega \times (0, T_*].\) It follows from (1.7) that there exists a \(T_1 \in (0, T_*]\) such that (3.2) holds for \(T=T_1\).
Next, set
and
Then \(T^*_1 \ge T_1 >0.\) In particular, Proposition 3.1 together with continuity argument implies that (3.2) in fact holds on \((0, T^*).\) Thus,
provided that \(m_0 < \varepsilon _0\) as assumed.
Moreover, for any \(0<\tau <T \le T^*\) with T finite, one deduces from standard embedding that
Combining (3.53) and (3.91) gives for any \(0<\tau <T \le T^*\),
where one has used the standard embedding
Moreover, it follows from (3.2), (3.5), (3.87) and [17, Lemma 2.3] that
Thanks to (3.23) and (3.91), the standard arguments yield that
which together with (4.44.5) and (4.6) gives
Since \((\rho , \varvec{u})\) satisfies (2.33.38) with \(\textbf{F}= -\rho \varvec{u}_t - \rho \varvec{u}\cdot \nabla \varvec{u}- \textrm{div}(\nabla \varvec{d} \odot \nabla \varvec{d})\), we deduce from (3.1)\(_2\), (4.44.5), (4.6), (4.8) and (3.91) that
Now, we claim that
Otherwise, \(T^* < \infty .\) Proposition 3.1 implies that (3.3) holds at \(T=T^*\). It follows from (3.87), (3.79) and (3.21) that
satisfies
Therefore, one can take \((\rho ^*, \varvec{u}^*, \varvec{d}^*)\) as the initial data and apply Lemma 2.1 again to extend the local strong solution beyond \(T^*\). This contradicts the assumption of \(T^*\) in (4.2). Hence, \(T^*=\infty .\) We thus complete the proof of Theorem 1.1 since exponential decay of solution (1.11) follows directly from (3.92) and (3.54).
Data Availability
No datasets were generated or analyzed during the current study.
References
Chandrasekhar, S.: Liquid Crystals. Cambridge University Press, Cambridge (1977)
Cho, Y., Kim, H.: Unique solvability for the density-dependent Navier-Stokes equations. Nonlinear Anal. 59(4), 465–489 (2004)
Ding, S., Huang, J., Xia, F.: Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum. Filomat 27, 1247–1257 (2013)
Ericksen, J.: Hydrostatic theory of liquid crystal. Arch. Rational Mech. Anal. 9, 371–378 (1962)
Feireisl, E.: Dynamics of viscous compressible fluids. Oxford University Press, Oxford (2004)
De Gennes, P.G.: The Physics of Liquid Crystals. Oxford University Press, London (1974)
Gao, J., Tao, Q., Yao, Z.: Strong solutions to the density-dependent incompressible nematic liquid crystal flows. J. Differ. Eq. 260(4), 3691–3748 (2016)
He, C., Li, J., Lü, B.: Global well-posedness and exponential stability of 3D Navier-Stokes equations with density-dependent viscosity and vacuum in unbounded domains. Arch. Ration. Mech. Anal. 239(3), 1809–1835 (2021)
Hu, X., Wang, D.: Global solution to the three-dimensional incompressible flow of liquid crystals. Comm. Math. Phys. 296(3), 861–880 (2010)
Huang, X., Wang, Y.: Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity. J. Differ. Eq. 259(4), 1606–1627 (2015)
Leslie, F.: Theory of flow phenomenon in liquid crystals. In: The theory of liquid crystals, pp. 1–81. Academic Press, London (1979)
Li, J.: Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions. Nonlinear Anal. 99, 80–94 (2014)
Li, J.: Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow. Methods Appl. Anal. 22(2), 201–220 (2015)
Li, L., Liu, Q., Zhong, X.: Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum. Nonlinearity 30, 4062 (2017)
Li, Q., Wang, C.: Local well-posedness of nonhomogeneous incompressible liquid crystals model without compatibility condition. Nonlinear Anal. Real World Appl. 65, Paper No. 103474, 22 pp (2022)
Li, X.: Wang, Dehua, Global strong solution to the density-dependent incompressible flow of liquid crystals. Trans. Amer. Math. Soc. 367(4), 2301–2338 (2015)
Lions, P. L.: Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xiv+237 pp
Liu, Q., Liu, S., Tan, W.: Zhong X, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows. J. Differ. Equat. 261, 6521–6569 (2016)
Liu, S., Zhang, J.: Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete Contin. Dyn. Syst. Ser. B. 21, 2631–2648 (2016)
Liu, Y.: Global well-posedness of the 3D incompressible nematic liquid crystal flows with density-dependent viscosity coefficient. Math. Methods Appl. Sci. 43(9), 5985–6010 (2020)
Liu, Y.: Global existence and exponential decay of strong solutions to the 2D density-dependent nematic liquid crystal flows with vacuum. Taiwanese J. Math. 24(5), 1205–1228 (2020)
Liu, Y., Zhong, X.: On the Cauchy problem of 3D nonhomogeneous incompressible Nematic liquid crystal flows with vacuum. Commun. Pure Appl. Anal. 19, 5219–5238 (2020)
Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13, 115–162 (1959)
Lü, B., Song, S.: On local strong solutions to the three-dimensional nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum. Nonlinear Anal. Real World Appl. 46, 58–81 (2019)
M. Struwe. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. 4th edition. vol. 34. Springer-Verlag, Berlin, 2008. xx+302 pp
Wen, H., Ding, S.: Solutions of incompressible hydrodynamic flow of liquid crystals. Nonlinear Anal. Real World Appl. 12(3), 1510–1531 (2011)
Xia, Y., Zhu, M.: Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity. Commun. Math. Sci. 22(1), 257–283 (2024)
Yu, H., Zhang, P.: Global strong solutions to the incompressible Navier-Stokes equations with density-dependent viscosity. J. Math. Anal. Appl. 444(1), 690–699 (2016)
Zhang, J.: Global well-posedness for the incompressible Navier-Stokes equations with density-dependent viscosity coefficient. J. Differ. Eq. 259(5), 1722–1742 (2015)
Zhong, X.: A remark on global strong solution of two-dimensional inhomogeneous nematic liquid crystal flows in a bounded domain. Math. Nachr. 294(7), 1428–1443 (2021)
Zhou, L., Tang, C.: Global well-posedness to the 3D Cauchy problem of nonhomogeneous Navier-Stokes equations with density-dependent viscosity and large initial velocity. J. Math. Phys. 64(11), Paper No. 111509 (2023)
Acknowledgements
The authors would like to thank the referees and editor for all valuable and helpful comments on the manuscript.
Funding
This work of the first author is supported by National Natural Science Foundation of China (Grant Nos. 12001495, 12371246).
Author information
Authors and Affiliations
Contributions
Jieqiong Liu and Huanyuan Li wrote the main manuscript text. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interest regarding the publication of the paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, H., Liu, J. Global existence and exponential decay of strong solutions to the 3D nonhomogeneous nematic liquid crystal flows with density-dependent viscosity. Z. Angew. Math. Phys. 75, 182 (2024). https://doi.org/10.1007/s00033-024-02322-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-024-02322-8
Keywords
- Nonhomogeneous nematic liquid crystal flows
- Global strong solution
- Exponential decay
- Large initial velocity
- Vacuum