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

Abstract

AbstractIn this paper, we construct invariant measures and global-in-time solutions for a fractional Schrödinger equation with a Moser–Trudinger type nonlinearity $$\begin{aligned} i\partial _t u= (-\Delta )^{\alpha }u+ 2\beta u e^{\beta |u|^2} \qquad \text{ for }\qquad (x,t)\in \ M\times \mathbb {R}\end{aligned}$$ i ∂ t u = ( - Δ ) α u + 2 β u e β | u | 2 for ( x , t ) ∈ M × R on a compact Riemannian manifold M without boundary of dimension $$d\ge 2$$ 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 $$s>d/2$$ s > d / 2 or if $$s\le d/2$$ s ≤ d / 2 and $$s\le 1+\alpha $$ s ≤ 1 + α , there exists an invariant measure $$\mu ^{s}$$ μ s and a set $$\Sigma ^s \subset H^s$$ Σ s ⊂ H s containing arbitrarily large data such that $$\mu ^{s}(\Sigma ^s ) =1$$ μ s ( Σ s ) = 1 and that (E) is globally well-posed on $$\Sigma ^{s}$$ Σ s . In the case when $$s>d/2$$ s > d / 2 , we also obtain a logarithmic upper bound on the growth of the $$H^r$$ H r -norm of our solutions for $$r<s$$ 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.