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Probabilistic well-posedeness for the nonlinear Schr\"odinger equation on the $2d$ sphere I: positive regularities

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arxiv 2404.18229 v2 pith:O2MAEM72 submitted 2024-04-28 math.AP

Nicolas Burq , Nicolas Camps , Chenmin Sun , Nikolay Tzvetkov This is my paper · ORCID
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keywords equationmathbbnonlinearodingerprobabilisticregularityschrsphere
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abstract

We establish the probabilistic well-posedness of the nonlinear Schr\"odinger equation on the $2d$ sphere $\mathbb{S}^{2}$. The initial data are distributed according to Gaussian measures with typical regularity $H^{s}(\mathbb{S}^{2})$, for $s>0$. This level of regularity goes significantly beyond existing deterministic results, in a regime where the flow map cannot be extended uniformly continuously.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation

    math.AP 2025-09 conditional novelty 7.0 of 10

    Renormalized BBM with rough Gaussian initial data converges in law to stochastic BBM forced by derivative of spatial white noise, for all regularities alpha <= 1/4.

  2. On probabilistic ill-posedness

    math.AP 2026-07 accept novelty 6.0 of 10

    The paper defines enhanced probabilistic well-posedness by imposing stability at the origin and reinterprets recent 'beyond variance blowup' results for dispersive PDEs as probabilistic ill-posedness.

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