Longmou Moffo Filone Gilson - $H^1$- Global wellposedness of Nonlinear Schröndinger equation with exponential Non-linearity on compact surface

lundi 12 oct. 2026, 13:00 → 15:15 Europe/Paris

435 (UMPA, ENS de Lyon)

 In this work, we establish a probabilistic global theory (existence for all times, uniqueness and continuity) in $H^1$ for the NLS equation with a Moser-Trudinger  nonlinearity posed on compact surfaces. This equation is known to be the two dimensional counterpart to the classical energy-critical Schrödinger equations. The authors  Colliander, Ibrahim, Majdoub and Masmoudii  have identified a trichotomy around the criticality of this equation based on the size of the total energy. In particular, 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 obtained in the Work of Jean Baptiste and Monsaingeon but their setting does 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 probabilistic estimates are derived through the IID limit procedure . Beyond the difficulties related to the borderline nature of the context, the major challenge resides in the need to satisfy two features that tend to play against each other : obtaining both continuity property of the flow and large 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 supercritical regime issue, we show that a modified energy, with regularity similar to the original total energy, admits values as high as desired, suggesting that the constructed set of data contains supercritical ones.