Local well-posedness of the nonlinear Schrodinger equation with a quadratic nonlinearity on the two-dimensional torus

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• Ruoyuan Liu, University of Edinburgh
• Monday 20 February 2023, 15:00-16:00
• WATN-LT C (G24).

If you have a question about this talk, please contact Yuzhao Wang.

In this talk, I will present results on local well-posedness of the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity, posed on the two-dimensional torus, from both deterministic and probabilistic points of view.

For the deterministic well-posedness, Bourgain (1993) proved local well-posedness of the quadratic NLS in H^s for any s > 0. In this talk, I will go over local well-posedness in L2, thus resolving an open problem of 30 years since Bourgain (1993). In terms of ill-posedness in negative Sobolev spaces, this result is sharp. As a corollary, a multilinear version of the conjectural L3-Strichartz estimate on the two-dimensional torus is obtained.

For the probabilistic well-posedness, I will talk about almost sure local well-posedness of the quadratic NLS with random initial data distributed according to a fractional derivative of the Gaussian free field, and also a certain probabilistic ill-posedness result when the initial data becomes very rough. In particular, this part of the talk shows that the prediction on the critical regularity made by the probabilistic scaling due to Deng, Nahmod, and Yue (2019) breaks down for this model.

The first part of the talk is based on a joint work with Tadahiro Oh (The University of Edinburgh).

This talk is part of the Analysis Seminar series.

This talk is included in these lists:

• Analysis Seminar
• School of Mathematics Events
• WATN-LT C (G24)

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