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Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations

Hiroyuki Hirayama, Shinya Kinoshita

Abstract

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space $H^{s}$. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in $H^s$ for $s\ge 1/2$ if $d=2$, and $s>1/2$ if $d=3$.

Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations

Abstract

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension and . This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space . But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in for if , and if .

Paper Structure

This paper contains 6 sections, 14 theorems, 124 equations, 1 table.

Table of Contents

  1. Introduction
  2. Bilinear estimates
  3. The estimates for low modulation, $2$D
  4. The estimates for low modulation, $3$D
  5. Proof of Proposition \ref{['key_be']}
  6. Proof of the well-posedness

Key Result

Theorem 1.1

We assume that $\alpha$, $\beta$, $\gamma \in {\mathbb R}\backslash \{0\}$ satisfy $\theta <0$ and $\kappa \ne 0$. (i) Let $d=2$ and $s\ge \frac{1}{2}$. Then, (NLS_sys) is locally well-posed in $H^{s}$. More precisely, for any $r>0$ and for all initial data $(u_{0}, v_{0}, w_{0})\in B_{r}(H^{s}\time of the system (NLS_sys) on $[0, T]$. Such solution is unique in $B_R(X^{s,\frac{1}{2},1}_{\alpha,T}

Theorems & Definitions (27)

  • Definition 1
  • Remark 1.1
  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2: Strichartz estimate
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 17 more