Short-time existence and uniqueness for two-phase harmonic map heat flow with a prescribed moving interface
Abstract
Let N+ and N- be disjoint compact smooth submanifolds of a Euclidean space, and let Λ⊂ N+× N- be a compact smooth embedded submanifold prescribing admissible pairs of one-sided traces. Given a smooth family of separating hypersurfaces Γt⊂𝕋d, we study harmonic map heat flows into N± in the two moving phases, subject to (u+,u-)∈Λ, (−∂νtu+,∂νtu-) ⊥ T(u+,u-)Λ on Γt. For q>d+2, initial data in Wq2−2/q satisfying these conditions at t=0 generate a unique short-time strong solution in Wq2,1, which is smooth up to the moving interface for every positive time; smooth initial data satisfying the compatibility conditions of every order yield solutions that are smooth up to the initial corner. We double the paired map across Λ by normal reflection, which turns the interface conditions into a Dirichlet problem near the interface, and match this problem with the two bulk phases by an overlapping space-time Schwarz map inside a Schauder fixed-point argument. When the interface evolves independently by mean curvature, we also obtain a blow-up alternative in terms of its curvature and the two phase gradients. As an application, we prove local smooth solvability for the periodic matrix-valued sharp-interface system appearing in the convergence theory of Fei, Lin, Wang, and Zhang (\emph{Invent. Math.} 233 (2023), 1--80).
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BibTeX
@misc{wang2026shorttime,
title = {{Short-time existence and uniqueness for two-phase harmonic map heat flow with a prescribed moving interface}},
author = {Xingyu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07736},
eprint = {2610.07736},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07736},
}
RIS
TY - UNPB
TI - Short-time existence and uniqueness for two-phase harmonic map heat flow with a prescribed moving interface
AU - Xingyu Wang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07736
UR - https://arxiv.org/abs/2610.07736
AB - Let N^+ and N^- be disjoint compact smooth submanifolds of a Euclidean space, and let Λ⊂ N^+× N^- be a compact smooth embedded submanifold prescribing admissible pairs of one-sided traces. Given a smooth family of separating hypersurfaces Γ_t⊂𝕋^d, we study harmonic map heat flows into N^± in the two moving phases, subject to (u^+,u^-)∈Λ, (−∂_ν_tu^+,∂_ν_tu^-) ⊥ T_(u^+,u^-)Λ on Γ_t. For q>d+2, initial data in W_q^2−2/q satisfying these conditions at t=0 generate a unique short-time strong solution in W_q^2,1, which is smooth up to the moving interface for every positive time; smooth initial data satisfying the compatibility conditions of every order yield solutions that are smooth up to the initial corner. We double the paired map across Λ by normal reflection, which turns the interface conditions into a Dirichlet problem near the interface, and match this problem with the two bulk phases by an overlapping space-time Schwarz map inside a Schauder fixed-point argument. When the interface evolves independently by mean curvature, we also obtain a blow-up alternative in terms of its curvature and the two phase gradients. As an application, we prove local smooth solvability for the periodic matrix-valued sharp-interface system appearing in the convergence theory of Fei, Lin, Wang, and Zhang (\emph{Invent. Math.} 233 (2023), 1--80).
ER -