Invariant measures and global well-posedness for a fractional Schr\"odinger equation with Moser-Trudinger type nonlinearity

TL;DR

This paper constructs invariant measures and proves global well-posedness for a fractional Schrödinger equation with Moser-Trudinger nonlinearity on compact manifolds, extending previous results and providing new regularity-based bounds.

Contribution

It introduces a method to establish invariant measures and global solutions for fractional NLS with Moser-Trudinger nonlinearity, including high-regularity support and growth bounds.

Findings

Existence of invariant measures supported on large data sets.

Global well-posedness of fractional NLS on these sets.

Logarithmic bounds on solution norm growth for high regularity.

Abstract

In this paper, we construct invariant measures and global-in-time solutions for a fractional Schr\" odinger equation with a Moser-Trudinger type nonlinearity $$ i\partial_t u= (-\Delta)^{\alpha}u+ 2\beta u e^{\beta |u|^2},\qquad\mbox{for}\qquad(x,t)\in \ M\times \mathbb{R} $$ on a compact Riemannian manifold $M$ without boundary of dimension $d\geq 2$. To do so, we use the so-called Inviscid-Infinite-dimensional limits introduced by Sy ('19) and Sy and Yu ('21). More precisely, we show that if $s>d/2$ or if $s\leq d/2$ and $s\leq 1+\alpha$, there exists an invariant measure $\mu^{s}$ and a set $\Sigma^s \subset H^s$ containing arbitrarily large data such that $\mu^{s}(\Sigma^s ) =1$ and that the fractional NLS is globally well-posed on $\Sigma^{s}$. For strong regularities $s>d/2$ we also obtain a logarithmic upper bound on the growth of the $H^r$-norm of our solutions for $r<s$. This gives new examples of invariant measures supported in highly regular spaces in comparison with the Gibbs measure constructed by Robert ('21) for the same equation.