Finite time blow-up for the nematic liquid crystal flow in dimension two

Abstract

We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain Ω ⊂ R 2 . Given any k distinct points in the domain, we develop a new inner-outer gluing method to construct solutions which blow up exactly at those k points as t goes to a finite time T . Moreover, we obtain a precise description of the blow-up.

Modeling and Computation of Liquid Crystals

“…For the simplified equation, Huang, Lin, Liu and Wang (2016) constructed solutions in a 3-D bounded domain with Dirichlet boundary data where the direction field blows up at finite time while the velocity field remains smooth. Lai, Lin, Wang, Wei and Zhou (2019) proves that, for any given set of points in R 2 , one can construct solutions with smooth initial data which blow up exactly at these points in a small time.…”

From the Landau-de Gennes theory to the Ericksen-Leslie theory in dimension two

“…Moreover, in dimension three, [43] has shown the connection between the solutions to the Q-tensor flows and the weak solutions to the harmonic map flows which may contain singular points. However, nontrivial singular weak solutions (with non-trivial velocity) to the Ericksen-Leslie equations do exist, see [18] for three-dimensional case and [22] for two-dimensional case. Our main goal in this paper is to study the connections between the solutions to the Beris-Edwards model and the weak solutions to the Ericksen-Leslie equations in dimension two.…”

“…In fact, two examples of weak solutions of finite time singularity in dimension three have been constructed in [18]. Recently, weak solutions with finite time singularities in dimension two have been constructed in [22]. For the blow-up criteria of strong solutions to the Ericksen-Leslie system, we refer to [19,15] and the references therein.…”

“…Indeed, this convergence has been shown in [14,15] on the time interval where the solution to (1.27) remains regular, and the methods in [14,15] depend crucially on the regularity of the strong solutions to (1.27). However, the extension of this approach in [14,15] to larger times seems impossible due to the existence of singular weak solutions to the Ericksen-Leslie equations [18,22]. In this respect, Kortum [20] proved the convergence of the weak solutions to (1.28) to the global-in-time weak solutions to (1.27) in two dimensional torus T 2 .…”

“…As mentioned in Remark 1.2, the asymptotic convergence of solutions to the Beris-Edwards system (1.3)-(1.5) to the regular solutions to the Ericksen-Leslie system (1.18)-(1.20) with the the coefficients satisfying (1.29)-(1.32) has been proved in [45]. However, the analysis in [45] is based on the Hilbert expansion, which depends crucially on the high order differentiability of the limiting solutions to the Ericksen-Leslie system and thus cannot be applied to the case that the solutions to the Ericksen-Leslie system have singularities whose existence had been confirmed in [18,22]. Here we will establish the asymptotic convergence of these two systems as ǫ → 0 + for weak solutions by analysing the a priori energy inequality (1.15) for (v ǫ , Q ǫ ) and showing that the weak-limit (v * , Q * ) of (v ǫ , Q ǫ ) solves the Ericksen-Leslie system (1.34) subject to the initial data (1.36) and satisfies the energy inequalities (1.37).…”