A new p-harmonic map flow with Struwe monotonicity

We construct and analyze solutions to a regularized homogeneous p-harmonic map flow equation for general \(p \ge 2\) $p\ge 2$ . The homogeneous version of the problem is new in the supercritical range \(2< p < n\) $2<p<n$ and features a monotonicity formula extending the one found by Struwe for \(p = 2\) $p=2$ . Such a formula is not available for the nonhomogeneous equation, and not needed for the homogeneous equation with \(p \ge n\) $p\ge n$ , but is crucial in the range under consideration. The main innovations of this paper are in reproducing the tools of the \(p = 2\) $p=2$ case in the qualitatively distinct setting of a non-divergence, quasilinear equation. In particular: the aforementioned monotonicity formula, a nonlinear Bochner formula, an appropriate \(\varepsilon \) $\epsilon$ -regularity result, and a Hessian bound valid in \(\varepsilon \) $\epsilon$ -regularity regions. The construction itself follows the general Ginzburg–Landau-type approximation strategy of Chen and Struwe, and we thereby obtain a strict generalization of the \(p = 2\) $p=2$ case: 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 system presents new obstacles that do not appear in the classical case \(p = 2\) $p=2$ and that are unrelated to the geometric nature of the equation, namely uniform-time existence for the approximating problem. Thus our basic existence result is stated conditionally.