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Probabilistic global-wellposedness for the energy-supercritical Schrödinger equations on compact manifolds

Seynabou Gueye, Filone G. Longmou-Moffo, Mouhamadou Sy

TL;DR

This work addresses the energy-supercritical nonlinear Schrödinger equation on compact manifolds by constructing invariant measures supported on Sobolev spaces $H^s$ with $s\le\frac{d}{2}$ and proving almost sure global well-posedness with polynomial-in-time growth bounds. The authors develop an IID-limit framework that blends fluctuation-dissipation ideas with Gibbs-type measures to build a statistical ensemble and an invariant measure $\mu$, extending globalization beyond deterministic thresholds. The approach yields a global flow on a full-measure set $\Sigma$ for general compact manifolds, with sharper results on the torus and Zoll manifolds, and provides a rigorous mechanism for long-time control via probabilistic invariants. The results advance understanding of supercritical dispersive dynamics on curved spaces and offer a robust pathway to global regularity in a probabilistic sense, including invariance and recurrence properties of the constructed ensemble.

Abstract

We consider the nonlinear Schrödinger 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.

Probabilistic global-wellposedness for the energy-supercritical Schrödinger equations on compact manifolds

TL;DR

This work addresses the energy-supercritical nonlinear Schrödinger equation on compact manifolds by constructing invariant measures supported on Sobolev spaces with and proving almost sure global well-posedness with polynomial-in-time growth bounds. The authors develop an IID-limit framework that blends fluctuation-dissipation ideas with Gibbs-type measures to build a statistical ensemble and an invariant measure , extending globalization beyond deterministic thresholds. The approach yields a global flow on a full-measure set for general compact manifolds, with sharper results on the torus and Zoll manifolds, and provides a rigorous mechanism for long-time control via probabilistic invariants. The results advance understanding of supercritical dispersive dynamics on curved spaces and offer a robust pathway to global regularity in a probabilistic sense, including invariance and recurrence properties of the constructed ensemble.

Abstract

We consider the nonlinear Schrödinger equations with a general nonlinearity power in all dimensions. We construct invariant measures concentrated on Sobolev spaces of singular orders, . 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.

Paper Structure

This paper contains 24 sections, 22 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Background and earlier results
  4. Local wellposedness
  5. Global well posedness
  6. Long-time behavior
  7. Probabilistic approaches to the global regularity problem
  8. Problematic and exposition of the main result
  9. Organization of the paper
  10. Acknowledgement
  11. General notations
  12. Local wellposedness
  13. Local well posedness
  14. Deterministic devices for the globalization
  15. Dissipation models and apriori estimates
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(M^d,g)$ be a compact Riemannian manifold of dimension $d\geq 3$. Let $s\in (s_{M^d},\frac{d}{2}]$ and $q\geq q_{M^d}$. For all $\epsilon>0,$ there are a set $\Sigma=\Sigma_{q,s,\epsilon}\subset H^s$ and a probability measure $\mu=\mu_{q,s,\epsilon}$ such that :

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Proposition \ref{['local well posedness']}
  • Corollary 2.3: General compact Riemannian manifold $M^d$
  • proof
  • Corollary 2.4: Case of the Torus $\mathbb{T}^d, d\geq 3$
  • ...and 36 more