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 $Λ\subset N^+\times N^-$ be a compact smooth embedded submanifold prescribing admissible pairs of one-sided traces. Given a smooth family of separating hypersurfaces $Γ_t\subset\mathbb T^d$, we study harmonic map heat flows into $N^\pm$ in the two moving phases, subject to \[ (u^+,u^-)\inΛ, \qquad \bigl(-\partial_{ν_t}u^+,\partial_{ν_t}u^-\bigr) \perp T_{(u^+,u^-)}Λ \quad\text{on }Γ_t. \] 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).