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arXiv · 2210.15066

Local well-posedness of the periodic nonlinear Schrödinger equation with a quadratic nonlinearity $\overline{u}^2$ in negative Sobolev spaces

Ruoyuan Liu
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Abstract

We study low regularity local well-posedness of the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $\overline{u}^2$, posed on one-dimensional and two-dimensional tori. While the relevant bilinear estimate with respect to the $X^{s, b}$-space is known to fail when the regularity $s$ is below some threshold value, we establish local well-posedness for such low regularity by introducing modifications on the $X^{s, b}$-space.

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Ruoyuan Liu. 2023-07-14. Local well-posedness of the periodic nonlinear Schrödinger equation with a quadratic nonlinearity $\overline{u}^2$ in negative Sobolev spaces. https://arxiv.org/abs/2210.15066

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