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arXiv · 2109.05626

Gibbs measure for the focusing fractional NLS on the torus

Rui Liang·Yuzhao Wang
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Abstract

We study the construction of the Gibbs measures for the {\it focusing} mass-critical fractional nonlinear Schr\"odinger equation on the multi-dimensional torus. We identify the sharp mass threshold for normalizability and non-normalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz-Rose-Speer (1988), Bourgain (1994), and Oh-Sosoe-Tolomeo (2021) on the one-dimensional nonlinear Schr\"odinger equations. To this purpose, we establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus, which is of independent interest.

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Rui Liang, Yuzhao Wang. 2021-09-12. Gibbs measure for the focusing fractional NLS on the torus. https://doi.org/10.1137/21m1445946

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