2025/08/26 by Dong, Huan, Ren, Siqi, Wang, Wei
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.18800
In this paper, we consider the isotropic-nematic phase transition with anisotropic elasticity governed by the Landau-de Gennes dynamics of liquid crystals. For -(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-(1)/(3)I) with n∈ \mathbbS2 and s+>0, where the alignment vector field n satisfies (2s+2∂t n+h)× n=0 and h=-(δE(n,∇ n))/(δn) with E(n,∇ n) denoting the Oseen-Frank energy; On the interface, the strong anchoring condition n=ν 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.