Probabilistic global-wellposedness for the energy-supercritical Schr\"odinger equations on compact manifolds

TL;DR

This paper establishes almost sure global well-posedness for energy-supercritical nonlinear Schrödinger equations on compact manifolds using invariant measures, extending results to singular Sobolev spaces and specific geometries.

Contribution

It introduces a probabilistic approach to prove global well-posedness for supercritical NLS on compact manifolds, including the torus and Zoll manifolds, with invariant measures on singular Sobolev spaces.

Findings

Almost sure global well-posedness established

Invariant measures constructed on Sobolev spaces of singular order

Results apply to general nonlinearities and specific manifold cases

Abstract

We consider the nonlinear Schr\"odinger equations with a general nonlinearity power in all dimensions. We construct invariant measures concentrated on Sobolev spaces $H^s$ of singular orders, $s\leq\frac{d}{2}$. We prove almost sure global wellposedness and bounds on the growth in time of the solutions via invariant measure arguments. Our setting includes a generic compact Riemannian manifold; we specify the cases of the torus and Zoll manifolds.