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SUMMARY:Longmou Moffo Filone Gilson - $H^1$- Global wellposedness of Nonli
 near Schröndinger equation with exponential Non-linearity on compact surf
 ace
DTSTART:20261012T110000Z
DTEND:20261012T131500Z
DTSTAMP:20261014T102100Z
UID:indico-event-17535@indico.math.cnrs.fr
DESCRIPTION: In this work\, we establish a probabilistic global theory (e
 xistence for all times\, uniqueness and continuity) in $H^1$ for the NLS e
 quation with a Moser-Trudinger  nonlinearity posed on compact surfaces. T
 his equation is known to be the two dimensional counterpart to the classic
 al energy-critical Schrödinger equations. The authors  Colliander\, Ibra
 him\, Majdoub and Masmoudii  have identified a trichotomy around the crit
 icality of this equation based on the size of the total energy. In particu
 lar\, for supercritical regimes (large energy)\, the equation is known to 
 exhibit instabilities : the (uniform) continuity of the flow fails to hold
 . Large data distributional non unique probabilistic solutions have been o
 btained in the Work of Jean Baptiste and Monsaingeon but their setting doe
 s not handle  the uniqueness issue for the $H^1$-data and therefore could
  not define a flow for this regularity. Our main focus here is to build a 
 single probabilistic framework that provides both existence\, uniqueness\,
  and continuity with respect to the initial data in $H^1$. Our uniqueness 
 and continuity are based on the so-called Yudovich argument \, and the pro
 babilistic estimates are derived through the IID limit procedure . Beyond 
 the difficulties related to the borderline nature of the context\, the maj
 or challenge resides in the need to satisfy two features that tend to play
  against each other : obtaining both continuity property of the flow and l
 arge data in the support of the reference measure. This made the design of
  the dissipation operator inherent in the method\, as well as the analysis
  of the resulting quantities\, particularly difficult. Regarding the super
 critical regime issue\, we show that a modified energy\, with regularity s
 imilar to the original total energy\, admits values as high as desired\, s
 uggesting that the constructed set of data contains supercritical ones.\n\
 nhttps://indico.math.cnrs.fr/event/17535/
LOCATION:435 (UMPA\, ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/17535/
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