Simultaneous Sharp-Interface Limits for a Diffuse-Interface Landau--de Gennes Droplet Model
Abstract
Nematic droplet models based on diffuse interfaces involve competing microscopic transition layers whose interaction is lost if the small parameters are sent to zero successively. We study this interaction for the diffuse-interface Landau--de Gennes model of Shi et al., with phase-field thickness ε and exterior Q-tensor relaxation length κε, and show that their relative scale selects the effective sharp-interface theory. Within a regular-interface class, we establish the corresponding sharp-interface limit for every limiting ratio r=limε→0κ/ε∈[0,+∞]. Three qualitatively different regimes emerge. When r=+∞, the two transition layers separate asymptotically and the interfacial costs decouple, recovering the previously obtained iterated limit. When r∈(0,+∞), the layers overlap and produce a genuinely coupled, r-dependent anisotropic surface density. When r=0, finite energy forces the interior trace of Q to vanish on the limiting interface. The comparable-scale regime requires a genuinely coupled interfacial analysis. Thus the exterior relaxation length is not merely a regularization parameter: together with the phase-field thickness, it determines the macroscopic surface energy and anchoring law of the sharp-interface limit.
Related
- Local Well-posedness of a Diffuse Interface Two-phase Flow Model with Surfactants
- Singular limits for general energy-critical complex Ginzburg-Landau equations
- Beyond the reduced Landau--Lifshitz--Bloch equation: Singular potential, global well-posedness, and micromagnetic limits
- Well-posedness of Strong Solutions for a Thermodynamically Consistent Diffuse-interface Model for Incompressible Two-phase Flows with a Soluble Surfactant
- A Robust All-Mach Six-Equation Diffuse-Interface Method for Multiphase Flows with Surface Tension
BibTeX
@misc{guan2026simultaneous,
title = {{Simultaneous Sharp-Interface Limits for a Diffuse-Interface Landau--de Gennes Droplet Model}},
author = {Sisi Guan and Wei Wang and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26440},
eprint = {2610.26440},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26440},
}
RIS
TY - UNPB
TI - Simultaneous Sharp-Interface Limits for a Diffuse-Interface Landau--de Gennes Droplet Model
AU - Sisi Guan
AU - Wei Wang
AU - Lei Zhang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26440
UR - https://arxiv.org/abs/2610.26440
AB - Nematic droplet models based on diffuse interfaces involve competing microscopic transition layers whose interaction is lost if the small parameters are sent to zero successively. We study this interaction for the diffuse-interface Landau--de Gennes model of Shi et al., with phase-field thickness ε and exterior Q-tensor relaxation length κ_ε, and show that their relative scale selects the effective sharp-interface theory. Within a regular-interface class, we establish the corresponding sharp-interface limit for every limiting ratio r=lim_ε→0κ/ε∈[0,+∞]. Three qualitatively different regimes emerge. When r=+∞, the two transition layers separate asymptotically and the interfacial costs decouple, recovering the previously obtained iterated limit. When r∈(0,+∞), the layers overlap and produce a genuinely coupled, r-dependent anisotropic surface density. When r=0, finite energy forces the interior trace of Q to vanish on the limiting interface. The comparable-scale regime requires a genuinely coupled interfacial analysis. Thus the exterior relaxation length is not merely a regularization parameter: together with the phase-field thickness, it determines the macroscopic surface energy and anchoring law of the sharp-interface limit.
ER -