Phase transition of an anisotropic Ginzburg-Landau equation (Q6574276)

In the study of liquid crystals, one often encounters elastic energies with anisotropy, i.e., energies with distinct coefficients multiplying the square of the divergence and the curl of the order parameters. Typical examples involve the Oseen-Frank model, Ericksen's model and the Landau-De Gennes model. From a microscopic point of view, the anisotropy of these models can be interpreted as excluded volume potential of molecular interaction. Anisotropic models also arise in the theory of superconductivity. The anisotropy brings various new challenges to the investigation of the variational problems and their gradient flows for the aforementioned models. Taking this into account, from the physical point of view, the aim of the present paper is to study an anisotropic system modeling the isotropic-nematic phase transition of a liquid crystal droplet. The author considers the anisotropic Ginzburg-Landau type energy. From the mathematical point of view, the effective geometric motions of an anisotropic Ginzburg-Landau equation with a small parameter \(\epsilon>0\), which characterizes the width of the transition layer, are studied. For a well-prepared initial datum, it is proved that as \(\epsilon\) tends to zero, the solutions develop a sharp interface limit which evolves under mean curvature flow. The bulk limits of the solutions correspond to a vector field \(u(x, t)\) which is of unit length on one side of the interface, and is zero on the other side. The main tools are a combination of the modulated energy method and weak convergence methods. In particular, by a boundary blow-up argument, it is shown that \(u\) must be tangent to the sharp interface. Moreover, it solves a geometric evolution equation for the Oseen-Frank model in liquid crystals.