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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
##### Keywords:
Lojasiewicz exponent
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