Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows

Pullback attractors of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows

“…In this paper, we consider the existence of pullback attractors of the following two-dimensional nonautonomous simplified Ericksen-Leslie system for nematic liquid crystal flows (see [1,19,20]) ∂u ∂t − νΔu + (u · ∇)u + ∇p = −(∇d) t Δd + g(x, t), (x, t) ∈ Ω × R τ , (1.1)…”

“…In this paper, we consider the long-time behaviour of a two-dimensional simplified Ericksen-Leslie system for nematic liquid crystal flows with two non-autonomous terms: one volume force on the linear momentum equation g(x, t) and the non-homogeneous Dirichlet boundary condition for the order parameter field h(x, t) (see [1,20]). …”

“…Therefore, it is interesting to study the existence of attractors of the Ericksen-Leslie system for nematic liquid crystal flows to explore the complexity of their geometric structure. There have been many works about the well-posedness of solutions and the long-time behaviour of solutions of the Ericksen-Leslie system for nematic liquid crystal flows(see [1,4,10,11,19,25,29,34]). For example, in [10], the authors proved the existence of trajectory attractors of the Sun-Liu model for nematic liquid crystals in 3D.…”

“…In [25], the authors considered the long-time behaviour of some mathematical models for nematic liquid crystals and proved any solution trajectory will converge to an equilibrium point and provided its convergence rate. The author in [1] considered the well-posedness of the weak solutions of the non-autonomous simplified Ericksen-Leslie system for nematic liquid crystal flows with one volume force on the linear momentum equation g(x, t) and the non-homogeneous Dirichlet boundary condition for the order parameter field h(x, t) by use of the semi-Galerkin method. Moreover, to guarantee the compactness of the symbol space, the stronger assumptions that g(x, t) and h(x, t) were L 2 translation bounded than that in this paper were added.…”

“…First, as the non-autonomous simplified Ericksen-Leslie system for nematic liquid crystal flows is supplied with the non-homogeneous Dirichlet boundary condition for the order parameter field h(x, t), we cannot choose −Δ(Δd − f (d)) as a test function to prove the existence of a pullback absorbing set in V × H 2 (Ω)(the definition of H and V will be given in the next section). To solve this problem, we use the same method as in [1] to obtain the existence of a pullback absorbing set in V × H 2 (Ω). Second, we cannot obtain the existence of a pullback absorbing set in more regular space than V × H 2 (Ω) to prove some appropriate compactness of the process {U (t, τ )} t≥τ .…”