Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data

Hiroyuki Hirayama

doi:10.3934/cpaa.2014.13.1563

Abstract

In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of the solution of the system in the Sobolev space $H^{s}$ for $s>d/2+3$ is proved by M. Colin and T. Colin. We prove the well-posedness of the system with low regularity initial data. For some cases, we also prove the well-posedness and the scattering at the scaling critical regularity by using $U^{2}$ space and $V^{2}$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).

Highlights

We consider the Cauchy problem of the system of Schrodinger equations: ((ii∂∂tt + + α∆)u β∆)v = = −(∇ −(∇ · · w)v, w)u,(t, x) ∈ (0, ∞) × Rd (t, x) ∈ (0, ∞) × Rd ((iu∂(t0+, xγ),∆v)(w0,= ∇(u · x), w(0, v), x))(t, x) ∈ = (u0(x)
The aim of this paper is to prove the well-posedness and the scattering of (1.1) in the scaling critical Sobolev space
As the known result for (1.1), we introduce the work by Colin and Colin ([6]). They proved that the local existence of the solution of (1.1) for s > d/2 + 3

Summary

Introduction

We consider the Cauchy problem of the system of Schrodinger equations:. where α, β, γ ∈ R\{0} and the unknown functions u, v, w are d-dimensional complex vector valued. We consider the Cauchy problem of the system of Schrodinger equations:. Where α, β, γ ∈ R\{0} and the unknown functions u, v, w are d-dimensional complex vector valued. The system (1.1) was introduced by Colin and Colin in [6] as a model of laser-plasma interaction. (1.1) is invariant under the following scaling transformation: Aλ(t, x) = λ−1A(λ−2t, λ−1x) (A = (u, v, w)),. The scaling critical regularity is sc = d/2 − 1. The aim of this paper is to prove the well-posedness and the scattering of (1.1) in the scaling critical Sobolev space. Schrodinger equation, well-posedness, Cauchy problem, scaling critical, Bilinear estimate, bounded p-variation. Schrodinger equation, well-posedness, Cauchy problem, scaling critical, Bilinear estimate, bounded p-variation. 1

HIRAYAMA

Bilinear Strichartz estimates

R R θ1 b θ2 b

Time local estimates

Proof of the well-posedness and the scattering

A priori estimates