Invariant measures and global well-posedness for a fractional Schrödinger equation with Moser-Trudinger type nonlinearity

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 $$ i\partial_t u= (-Δ)^αu+ 2βu e^{β|u|^2},\qquad\mbox{for}\qquad(x,t)\in \ M\times \mathbb{R} $$ on a compact Riemannian manifold $M$ without boundary of dimension $d\ge 2$. 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 $s\le d/2$ and $s\le 1+\alpha$, 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.