partial differential equations

Existence of weak solutions for two-phase matrix-valued harmonic map flows

arXiv:2506.16278 · doi:10.1016/j.jfa.2026.111623

summary

The paper proves the existence of weak solutions for a two‑phase matrix‑valued harmonic map flow, using a time‑discretized minimizing‑movement scheme linked to a prescribed interface.

Abstract

We study the existence of weak solutions to a two-phase matrix-valued harmonic map flow with lifespan determined by that of the prescribed interface. This system arises as the limiting equation for the matrix-valued Keller--Rubinstein--Sternberg problem studied in (Invent. Math., 233(1):1--80, 2023). Our construction is based on a modified minimizing movement scheme. Precisely, we discretize time and build approximate solutions by interpolating minimizers of the associated variational problem on each time step.

35 pages. Final version accepted for publication. Several unclear points in the previous version have been clarified

Topics & keywords

#matrix-valued harmonic map flow#weak solutions#two-phase flow#variational methods#minimizing movement schemeweak solutionharmonic map flowmatrix-valuedminimizing movementKeller–Rubinstein–Sternberginterface dynamics
Existence of weak solutions for two-phase matrix-valued harmonic map flows · wovepaper