A new $p$-harmonic map flow with Struwe monotonicity

TL;DR

This paper introduces a new homogeneous p-harmonic map flow with a Struwe-type monotonicity formula, constructing solutions via a Ginzburg-Landau approximation and analyzing their convergence and regularity properties.

Contribution

It develops the first homogeneous p-harmonic map flow with a monotonicity formula extending Struwe's for p=2, using novel approximation and regularity techniques.

Findings

Established a monotonicity formula for the homogeneous p-harmonic map flow.

Proved subsequential convergence of approximations away from a concentration set.

Identified new challenges in existence theory due to the quasilinear, non-divergence structure.

Abstract

We construct and analyze solutions to a regularized homogeneous $p$-harmonic map flow equation for general $p \geq 2$. The homogeneous version of the problem is new and features a monotonicity formula extending the one found by Struwe for $p = 2$; such a formula is not available for the nonhomogeneous equation. The construction itself is via a Ginzburg-Landau-type approximation \`a la Chen-Struwe, employing tools such as a Bochner-type formula and an $\varepsilon$-regularity theorem. We similarly obtain strong subsequential convergence of the approximations away from a concentration set with parabolic codimension at least $p$. However, the quasilinear and non-divergence nature of the equation presents new obstacles that do not appear in the classical case $p = 2$, namely uniform-time existence for the approximating problem, and thus our basic existence result is stated conditionally.