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Global Dynamics of the 2d NLS with White Noise Potential and Generic Polynomial Nonlinearity

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Abstract

Using an approach introduced by Hairer–Labbé we construct a unique global dynamics for the NLS on \({\mathbb {T}}^2\) with a white noise potential and an arbitrary polynomial nonlinearity. We build the solutions as a limit of classical solutions (up to a phase shift) of the same equation with smoothed potentials. This is an improvement on previous contributions of us and Debussche–Weber dealing with quartic nonlinearities and cubic nonlinearities respectively. The main new ingredient are space–time estimates for the approximate nonlinear solutions exploiting the time averaging effect for dispersive equations (the previous works were based only on fixed time spatial estimates).

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Acknowledgements

We warmly thank the anonymous referees for the detailed and constructive remarks on the manuscript.

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Authors and Affiliations

  1. UMR 5669 du CNRS, ENS Lyon, UMPA, 69007, Lyon, France

    N. Tzvetkov

  2. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56100, Pisa, Italy

    N. Visciglia

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  1. N. Tzvetkov
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  2. N. Visciglia
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Correspondence to N. Tzvetkov.

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Communicated by M. Hairer.

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N.T. is supported by ANR grant ODA (ANR-18-CE40-0020-01), N.V. is supported by the project PRIN Grant 2020XB3EFL and he acknowledge the Gruppo Nazionale per l’ Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituzione Nazionale di Alta Matematica (INDAM).

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Tzvetkov, N., Visciglia, N. Global Dynamics of the 2d NLS with White Noise Potential and Generic Polynomial Nonlinearity. Commun. Math. Phys. 401, 3109–3121 (2023). https://doi.org/10.1007/s00220-023-04707-8

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  • DOI: https://doi.org/10.1007/s00220-023-04707-8

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