In this work, we study the nematic–isotropic phase transition based on the dynamics of the Landau–De Gennes theory of liquid crystals. At the critical temperature, the Landau–De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.

在这项工作中，我们基于Landau-De Gennes 液晶理论的动力学研究了向列-各向同性相变。在临界温度下，Landau-De Gennes 体势同样有利于各向同性相和向列相。当弹性系数远小于体势的弹性系数时，可以通过形式渐近展开推导出标度极限：解梯度集中在由平均曲率流演化的封闭表面上。此外，在表面的一侧，溶液趋向于向列相，该相受谐波映射热流控制进入球体，而在另一侧，它趋向于各向同性相。为了严格证明这种缩放限制，我们通过结合弱收敛方法和调制能量方法来证明收敛结果。