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VOL. 85 | 2020 Random data Cauchy problem for the quadratic nonlinear Schrödinger equation without gauge invariance
Mamoru Okamoto

Editor(s) Yoshikazu Giga, Nao Hamamuki, Hideo Kubo, Hirotoshi Kuroda, Tohru Ozawa

Abstract

This article is concerned with the Cauchy problem for the quadratic nonlinear Schrödinger equation without gauge invariance. By randomizing the initial data, we prove that the Cauchy problem is almost surely locally well-posed in $H^s(\mathbb{R}^d)$ for $d \ge 5$ and $\frac{d-4}{d-3}s_c \lt s \lt s_c$, where $s_c := \frac{d}{2} - 2$ is the scaling critical regularity.

Information

Published: 1 January 2020
First available in Project Euclid: 29 December 2020

Digital Object Identifier: 10.2969/aspm/08510337

Subjects:
Primary: 35Q55

Rights: Copyright © 2020 Mathematical Society of Japan

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