NLS with exponential nonlinearity on compact surfaces

In this paper, we establish a probabilistic global theory in $H^1$ for the NLS with a Moser-Trudinger nonlinearity posed on compact surfaces. This equation is known to be the two dimensional counterpart to the classical energy-critical Schr\"odinger equations \cite{CollianderIbrahimMajdoubMasmoudi2009}. The authors of \cite{CollianderIbrahimMajdoubMasmoudi2009} also identified a trichotomy around the criticality of the 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 \cite{CasterasMonsaingeon2024}. The setting of \cite{CasterasMonsaingeon2024} 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 Yudowich argument \cite{Judovic1963}, and the probabilistic estimates are derived through the IID limit procedure \cite{Sy2019}. 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, 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.