On the asymptotic limit for the dynamic isotropic-nematic phase transition with anisotropic elasticity

In this paper, we consider the isotropic-nematic phase transition with anisotropic elasticity governed by the Landau-de Gennes dynamics of liquid crystals. For $-\frac{3}{2}< L<0,$ we rigorously justify the limit from the Landau-de Gennes flow to a sharp interface system characterized by a two-phase flow: The interface evolves via motion by mean curvature; In the isotropic region, $Q=0$; In the nematic region, $Q=s_+(nn-\frac{1}{3}I)$ with $n\in \mathbb{S}^2$ and $s_+>0$, where the alignment vector field $n$ satisfies $$(2s_+^2\partial_t n+h)\times n=0$$ and $h=-\frac{\delta E(n,\nabla n)}{\delta n}$ with $E(n,\nabla n)$ denoting the Oseen-Frank energy; On the interface, the strong anchoring condition $n=\nu$ is satisfied. This result rigorously verifies a claim made by de Gennes [Mol. Cryst. Liq. Cryst. 1971] regarding the surface tension strength of isotropic-nematic interfaces in dynamical settings. Furthermore, we rigorously justify this limit using the method of matched asymptotic expansions. First, we employ the idea of ``quasi-minimal connecting orbits'' developed by Fei-Lin-Wang-Zhang [Invent.math. 2023] to construct approximated solutions up to arbitrary order. Second, we derive a uniform spectral lower bound for the linearized operator around the approximate solution. To achieve this, we introduce a suitable basis decomposition and a coordinate transformation to reduce the problem to spectral analysis of two scalar one-dimensional linear operators and some singular product estimates. To address the difficulties arising from anisotropic elasticity and the strong anchoring boundary condition, we introduce a div-curl decomposition and, when estimating the cross terms, combine these with the anisotropic elastic terms to close the energy estimates.