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Abstract. In this paper, we derive the sharp interface model of the nematic-isotropic phase transition from the Landau-de Gennes theory by using the matched asymptotic expansion method. The model includes the evolution equation of the velocity and director field of the liquid crystal, the sharp interface and Young-Laplace jump condition on the interface.

We study the existence and concentration of solutions to the N-dimensional nonlinear Schrödinger equation −ε 2 u ε + V (x)u ε = K (x)|u ε | p−1 u ε + Q(x)|u ε | q−1 u ε with u ε (x) > 0 and u ε ∈ H 1 ‫ޒ(‬ N), where N ≥ 3, 1 < q < p < (N+2)/(N−2), and ε > 0 is sufficiently small. We take potential functions V (x) ∈ C ∞ 0 ‫ޒ(‬ N) with V (x) ≡ 0 and V (x) ≥ 0, and show that if K (x) and Q(x) are permitted to be unbounded under some necessary restrictions, then a positive solution u ε (x) exists in H 1 ‫ޒ(‬ N) when the corresponding ground energy function G(x) has local minimum points. We establish the concentration property of u ε (x) as ε tends to zero. We have removed from some previous papers the crucial restriction that the nonnegative potential function V (x) has a positive lower bound or decays at infinity like (1 + |x|) −α with 0 < α ≤ 2.

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In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two‐phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by ϵ, and the mobility coefficient (the diffusional Peclet number) is ϵα. We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as ϵ→0 in the case of 0≤α < 1. Copyright © 2016 John Wiley & Sons, Ltd.