Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2512.21634 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908731916156928 |
|---|---|
| author | Huang, Jiaxi Tataru, Daniel |
| author_facet | Huang, Jiaxi Tataru, Daniel |
| contents | 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 Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger 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 Schrödinger 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21634 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local well-posedness of the skew mean curvature flow for large data Huang, Jiaxi Tataru, Daniel Analysis of PDEs 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 Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger 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 Schrödinger 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. |
| title | Local well-posedness of the skew mean curvature flow for large data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.21634 |