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Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties. (English) Zbl 1259.35040
The authors consider the following evolutionary system that models the dynamics of nematic liquid crystal flows, \[ \begin{cases} v_t + v \cdot \nabla v -v \Delta v + \nabla P =-\lambda \nabla \cdot [\nabla d \odot \nabla d \\ \quad + \alpha (\Delta d -f(d) ) \otimes d - (1-\alpha ) d \otimes (\Delta d -f(d))], \\ \nabla \cdot v =0 \\ d_t + (v \cdot \nabla )d-\alpha (\nabla v ) d + (1-\alpha )(\nabla ^T v ) d = \gamma (\Delta d -f(d)), \end{cases}\tag{1} \] in \(Q \times (0, \infty )\), where \(Q\) is a unit square in \({\mathbb R}^2\) or a unit box in \({\mathbb R}^3\). Here \(v\) is the velocity field of the flow and \(d\) represents the averaged macroscopic molecular orientations. \(P\) is a scalar function representing the hydrodynamic pressure. The constants \(\nu ,\lambda \) and \(\gamma \) stand for viscosity, the competition between kinetic energy and potential energy, and macroscopic elastic relaxation time for the molecular orientation field, respectively. The constant \(\alpha \in [0,1]\) is a parameter related to the shape of the liquid crystal molecule. The penalty function \(f(d)= \frac{1}{\eta ^2} (| d | ^2-1) d \) with \(\eta \in (0,1]\). They consider the equations (1) under the boundary condition \[ v(x+e_i ) = v(x), d(x+e_i)=d(x) \quad \mathrm{for}\, \, x \in {\mathbb R}^n {(2)} \] and the initial condition \[ v| _{t=0}= v_0(x), \quad \mathrm{with }\,\, \nabla \cdot v_0=0, d| _{t=0}=d_0(x), \quad \mathrm{for }\,\, x \in Q. {(3)} \] Define the function spaces \(H^m _p(Q)= \{v \in H^m({\mathbb R}^n , {\mathbb R}^n); v(x+e_i)= v(x)\}\), \(\dot{H}^m_p(Q)= H_p^m (Q) \cap \left\{ v; \int _Q v(x)dx=0\right \}\) and \(V= \{ v \in L^2 _p (Q,{\mathbb R}^n); \nabla \cdot v =0\}\) where \(L^2_p(Q,{\mathbb R}^n)=H_p^0 (Q)\). Then the authors get the following theorems.
Theorem 1 When \(n=2\), then for any initial data \((v_0,d_0)\in V \times H_p^2(Q)\) and \(\alpha \in [0,1]\), the problem (1)–(3) admits a unique global classical solution. It converges to a steady state \((0,d_{\infty })\) as \(t \to \infty \) such that \[ \lim _{t \to \infty } (\| v(t) \| _{H^1}+ \| d(t)-d_{\infty }\| _{H^2})=0, \] where \(d_{\infty }\) satisfies the following nonlinear elliptic periodic boundary value problem: \[ -\Delta d_{\infty }+f (d_{\infty })=0,\quad d_{\infty }(x+e_i)= d_{\infty }(x) \quad (x \in {\mathbb R}^2). \] Moreover, \[ \| v(t) \| _{H^1}+ \| d(t) -d_{\infty }\| _{H^2} \leq C (1+t)^{-\theta /(1-2\theta )}, \qquad \forall t \geq 0, \] where \(C\) is a positive constant depending on \(\nu , \| v_0\| _{H^1}, \| d_0 \| _{H^2}, \| d_{\infty }\| _{H^2}\). The constant \(\theta \in (0,1/2)\) is the so-called Lojasiewicz exponent depending on \(d_{\infty }\).
Theorem 2 When \(n=3\), then for any initial data \((v_0,d_0) \in V\times H_p^2(Q)\) and \(\alpha \in [0,1]\), if the constant viscosity is sufficiently large such that \(\nu \geq \nu _0(v_0,d_0)\), then the problem (1)–(3) admits a unique global classical solution enjoying the same long-time behavior as in Theorem 1.
Theorem 3 When \(n=3\), let \(d^* \in H_p^2(Q)\) be an absolute minimizer of the functional \[ E(d)= \frac12 \| \nabla d \| ^ 2+ \int _Q F(d)dx. \] Then there exists a constant \(\sigma \in (0,1]\), which may depend on \(\nu , f, \) and \(d^*\) , such that for any \(\alpha \in [0,1]\) and initial data \((v_0,d_0) \in V \times H_p^2(Q)\) satisfying \(\| v_0\| _{H^1}+ \| d_0-d^* \| _{H^2}< \sigma \), the problem (1)–(3) admits a unique global classical solution enjoying the same long-time behavior as in Theorem 1.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76A15 Liquid crystals
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