Geodesic
The second Picard iteration of NLS on the $2d$ sphere does not regularize Gaussian random initial data
Nicolas Burq1 orcid ; Nicolas Camps2 orcid ; Mickaël Latocca3 orcid ; Chenmin Sun4 orcid ; Nikolay Tzvetkov5 orcid
1Université Paris-Saclay, Orsay, France
2Université de Rennes, Rennes, France
3Université d’Evry Val d’Essonne, Évry, France
4Université Paris-Est Créteil, Créteil, France
5École Normale Supérieure de Lyon, Lyon, France
EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 123-154

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DOI

We consider the Wick-ordered cubic Schrödinger equation (NLS) posed on the two-dimensional sphere, with initial data distributed according to a Gaussian measure. We show that the second Picard iteration does not improve the regularity of the initial data in the scale of the classical Sobolev spaces. This is in sharp contrast with the Wick-ordered NLS on the two-dimensional tori, a model for which we know from the work of Bourgain that the second Picard iteration gains one half-derivative. Our proof relies on identifying a singular part of the nonlinearity. We show that this singular part is responsible for a concentration phenomenon on a large circle (i.e., a stable closed geodesic), which prevents any regularization in the second Picard iteration.
DOI : 10.4171/emss/92
Classification : 35Q41
Mots-clés : Picard iteration

Nicolas Burq  1   ; Nicolas Camps  2   ; Mickaël Latocca  3   ; Chenmin Sun  4   ; Nikolay Tzvetkov  5

1 Université Paris-Saclay, Orsay, France
2 Université de Rennes, Rennes, France
3 Université d’Evry Val d’Essonne, Évry, France
4 Université Paris-Est Créteil, Créteil, France
5 École Normale Supérieure de Lyon, Lyon, France 
Nicolas Burq; Nicolas Camps; Mickaël Latocca; Chenmin Sun; Nikolay Tzvetkov. The second Picard iteration of NLS on the $2d$ sphere does not regularize Gaussian random initial data. EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 123-154. doi: 10.4171/emss/92
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     title = {The second {Picard} iteration of {NLS} on the $2d$ sphere does not regularize {Gaussian} random initial data},
     journal = {EMS surveys in mathematical sciences},
     pages = {123--154},
     year = {2025},
     volume = {12},
     number = {1},
     doi = {10.4171/emss/92},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/emss/92/}
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