Local well-posedness of the skew mean curvature flow for large data

TL;DR

This paper establishes local well-posedness for the skew mean curvature flow with large initial data in low-regularity Sobolev spaces, introducing new techniques for existence, uniqueness, and regularity propagation.

Contribution

It introduces novel methods such as time discretization, parallel transport frame construction, intrinsic fractional spaces, and gauge-independent difference equations for large data analysis.

Findings

Proved local well-posedness for large data in low-regularity Sobolev spaces.

Developed a robust framework for existence and uniqueness of solutions.

Introduced intrinsic fractional function spaces to control the flow.

Abstract

The skew mean curvature flow is an evolution equation for $d$ dimensional ma\-nifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schr\"odinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schr\"odinger Map equation. In this article, we prove large data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 2$. This is achieved by introducing several new ideas: (i) a time discretization method to establish the existence of smooth solutions, (ii) constructing the orthonormal frame by a parallel transport method and a lifting criterion, (iii) introducing intrinsic fractional function spaces $X^s\subset H^s$ on a noncompact manifold for any $s>\frac{d}{2}$, such that the $X^s$-norm of the second fundamental form can be propagated well along the quasilinear Schr\"odinger flow, (iv) deriving a difference equation to prove the uniqueness result for solutions $F\in C^2$, which is independent in the choices of gauge. Our method turns out to be more robust for large data problem.