Singularity and existence for a wave system of nematic liquid crystals

Abstract

Abstract. In this paper, we prove the global existence and singularity formation for a wave system from modelling nematic liquid crystals in one space dimension. In our model, although the viscous damping term is included, the solution with smooth initial data still has gradient blowup in general, even when the initial energy is arbitrarily small.

Poiseuille Flow of Nematic Liquid Crystals via the Full Ericksen–Leslie Model

“…As mentioned in Section 1.2, the two examples of finite time singularity formation constructed in [23] for the parabolic system over bounded regions are directly related to or caused by some non-trivial global/topological conditions. While as the singularity claimed in Theorem 1 is formed in essentially the same mechanism as that in [12,18] -it is created locally due to interactions of local waves that are of finite speed. A typical point singularity of direction field n of three dimensional parabolic system is in the form of x/|x|, which is not continuous at singular point.…”

“…Later this result was extended to more general initial data in [20], the case with damping in [12] and the variational wave system with n ∈ S 2 in [11,39,40]. Especially, in [12], the authors showed that behaviors of large solutions of the variational wave systems with and without damping are similar. The global well-posedness of Hölder continuous conservative solutions was established for the variational wave system, including: uniqueness [3,7], Lipschitz continuous dependence on some optimal transport metric [1], and generic regularity [2,5].…”

“…On the other hand, the existence of global energy conservative solutions after the singularity formation was established in [6]. Later this result was extended to more general initial data in [20], the case with damping in [12] and the variational wave system with n ∈ S 2 in [11,39,40]. Especially, in [12], the authors showed that behaviors of large solutions of the variational wave systems with and without damping are similar.…”

“…Solutions of the variational wave equation with smooth initial data could in general produce cusp singularities due to local interactions of waves; more precisely, there may be finite time blowup in their gradients while the solutions themselves are still Hölder continuous ( [10,12,18]). On the other hand, the existence of global energy conservative solutions after the singularity formation was established in [6].…”

“…Our method of showing the formation of singularity is thus based on those in papers [10,12,18] for a variational wave equation while there are several new ideas provided to cope with the new model. For example, we need understand the impact of the source term u x in (1.1) 2 .…”