Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

NLS with exponential nonlinearity on compact surfaces

Filone G. Longmou-Moffo, Mouhamadou Sy

TL;DR

This work develops a probabilistic global theory for the nonlinear Schrödinger equation with a Moser–Trudinger type exponential nonlinearity on compact 2D surfaces. It introduces a fluctuation–dissipation (IID) framework with finite-dimensional Galerkin approximations, a carefully designed dissipation operator, and an invariant-measure approach to overcome deterministic ill-posedness in the energy-supercritical regime. The authors prove existence, uniqueness, and continuity of the flow on the support of an invariant $H^1$-measure $\mu$, and show that the modified energy can attain arbitrarily large values, indicating the inclusion of supercritical data. The results extend global well-posedness and invariant-measure techniques to compact manifolds without restricting the nonlinearity parameter, offering a probabilistic route to long-time dynamics and stability for a borderline critical problem. The framework yields a well-defined random flow with an invariant law, enabling analysis of long-time behavior in a setting where deterministic theory struggles.

Abstract

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 Schrödinger 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.

NLS with exponential nonlinearity on compact surfaces

TL;DR

This work develops a probabilistic global theory for the nonlinear Schrödinger equation with a Moser–Trudinger type exponential nonlinearity on compact 2D surfaces. It introduces a fluctuation–dissipation (IID) framework with finite-dimensional Galerkin approximations, a carefully designed dissipation operator, and an invariant-measure approach to overcome deterministic ill-posedness in the energy-supercritical regime. The authors prove existence, uniqueness, and continuity of the flow on the support of an invariant -measure , and show that the modified energy can attain arbitrarily large values, indicating the inclusion of supercritical data. The results extend global well-posedness and invariant-measure techniques to compact manifolds without restricting the nonlinearity parameter, offering a probabilistic route to long-time dynamics and stability for a borderline critical problem. The framework yields a well-defined random flow with an invariant law, enabling analysis of long-time behavior in a setting where deterministic theory struggles.

Abstract

In this paper, we establish a probabilistic global theory in 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 Schrödinger 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 -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 . 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.
Paper Structure (22 sections, 11 theorems, 162 equations)

This paper contains 22 sections, 11 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Background and earliers results
  4. Yudowich argument for the uniqueness problem of the 2D Euler
  5. Problematic, Methodology and Exposition of the main result
  6. Difficulties
  7. General notations
  8. On a general framework of the Yudowich argument
  9. Stochastic dynamic and Probabilistic estimates
  10. Analysis of Dissipation
  11. Stochastic Global well-posedness
  12. The Markov semi-group and stationnary measure
  13. Inviscid Limit
  14. Infinite-dimensional limit and Global theory
  15. Existence of global solution
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $M$ be a compact surface endowed with a Riemannian metric, let $\beta>0, ~\delta\in(0,1)$ and $\gamma >4\beta$. There exist a probability space $(\Omega,\mathcal{F},\mathbb{P})$, a probability measure $\mu \in P(L^2)$ supported on $H^1$ and a stochastic process $u$ valued in $C(\mathbb{R},H^1)\c

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 18 more