Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schrödinger equation with angular regularity

2022/09/25 by Hiroyuki Hirayama, Hirayama, Hiroyuki, Shinya Kinoshita +3
Mathematics · #Advanced Mathematical Physics Problems #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2209.12176

Abstract

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in ℝ × ℝ² with the nonlinearity η|u|² where η∈ ℂ ∖ \0\ and low regularity initial data. If s < -1/4, the ill-posedness result in the Sobolev space H^s(ℝ²) is known. We will prove the well-posedness in H^s(ℝ²) for -1/2 < s < -1/4 by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces.

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