Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces

Hiroyuki Hirayama, Shinya Kinoshita, Mamoru Okamoto

Abstract

In this paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations introduced by Colin and Colin (2004). We determine an almost optimal Sobolev regularity where the smooth flow map of the Cauchy problem exists, expect for the scaling critical case. This result covers a gap left open in papers of the first and second authors (2014, 2019).

Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces

Abstract

In this paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations introduced by Colin and Colin (2004). We determine an almost optimal Sobolev regularity where the smooth flow map of the Cauchy problem exists, expect for the scaling critical case. This result covers a gap left open in papers of the first and second authors (2014, 2019).

Paper Structure

This paper contains 8 sections, 19 theorems, 166 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Fourier restriction norm spaces
  4. Linear and bilinear estimates
  5. Proof of bilinear estimates for $d=2$
  6. Proof of key propositions
  7. Proof of bilinear estimates for $d=3$
  8. The lack of the third differentiability of the flow map

Key Result

Theorem 1.1

Let $d=1$, $\mu >0$, and $s<0$. Then, the flow map of (NLS_sys) is not $C^3$ in $\mathcal{H}^s({\mathbb R}^d)$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Definition 1
  • Proposition 2.2
  • Remark 2.1
  • Proposition 2.3: Strichartz estimate (cf. GV85, KT98)
  • Corollary 2.4
  • Proposition 2.5
  • Corollary 2.6
  • ...and 22 more