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

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.