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

In this paper, we construct invariant measures and global-in-time solutions for a fractional Schrodinger equation with a Moser-Trudinger type nonlinearity i & part;(t)u = (-delta)(alpha)u + 2 beta ue(beta|u|2) for (x, t) is an element of M x R (E) on a compact Riemannian manifold M without boundary of dimension d >= 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 < d/2 and s <= 1 +alpha, there exists an invariant measure mu(s) and a set sigma(s) subset of H-s containing arbitrarily large data such that mu(s)(sigma(s)) = 1 and that (E) is globally well-posed on sigma(s). In the case when 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.