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To do so, we use the so-called Inviscid-Infinite-dimensional limits introduced by Sy (\u201919) and Sy and Yu (\u201921). More precisely, we show that if $$s>d\/2$$<\/jats:tex-math>\n \n s<\/mml:mi>\n ><\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or if $$s\\le d\/2$$<\/jats:tex-math>\n \n s<\/mml:mi>\n \u2264<\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$s\\le 1+\\alpha $$<\/jats:tex-math>\n \n s<\/mml:mi>\n \u2264<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \u03b1<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, there exists an invariant measure $$\\mu ^{s}$$<\/jats:tex-math>\n \n \u03bc<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and a set $$\\Sigma ^s \\subset H^s$$<\/jats:tex-math>\n \n \n \u03a3<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n \u2282<\/mml:mo>\n \n H<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> containing arbitrarily large data such that $$\\mu ^{s}(\\Sigma ^s ) =1$$<\/jats:tex-math>\n \n \n \u03bc<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n \u03a3<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and that (E) is globally well-posed on $$\\Sigma ^{s}$$<\/jats:tex-math>\n \n \u03a3<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In the case when $$s>d\/2$$<\/jats:tex-math>\n \n s<\/mml:mi>\n ><\/mml:mo>\n d<\/mml:mi>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we also obtain a logarithmic upper bound on the growth of the $$H^r$$<\/jats:tex-math>\n \n H<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-norm of our solutions for $$r<s$$<\/jats:tex-math>\n \n r<\/mml:mi>\n <<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This gives new examples of invariant measures supported in highly regular spaces in comparison with the Gibbs measure constructed by Robert (\u201921) for the same equation.<\/jats:p>","DOI":"10.1007\/s40072-023-00287-9","type":"journal-article","created":{"date-parts":[[2023,2,6]],"date-time":"2023-02-06T06:03:01Z","timestamp":1675663381000},"page":"416-465","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Invariant measures and global well-posedness for a fractional Schr\u00f6dinger equation with Moser-Trudinger type nonlinearity"],"prefix":"10.1007","volume":"12","author":[{"given":"Jean-Baptiste","family":"Casteras","sequence":"first","affiliation":[]},{"given":"L\u00e9onard","family":"Monsaingeon","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,2,6]]},"reference":[{"key":"287_CR1","volume-title":"Gradient Flows: In Metric Spaces and in the Space of Probability Measures","author":"L Ambrosio","year":"2008","unstructured":"Ambrosio, L., Gigli, N., Savar\u00e9, G.: Gradient Flows: In Metric Spaces and in the Space of Probability Measures. 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