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The Cauchy problem for a Schroedinger - Korteweg - de Vries system with rough data

Hartmut Pecher

TL;DR

This work establishes global well-posedness for a 1D Schrödinger–KdV system with rough initial data by combining refined bilinear Strichartz estimates with the I-method to create almost conserved energy functionals. The authors first secure local well-posedness in H^s for s>0 and then control the growth of modified energies E(Iu,Iv) and L(Iu,Iv) via detailed multilinear estimates and dyadic decompositions. By iterating short-time solutions and tuning the smoothing parameter N, they derive global results: s>3/5 when β=0 and s>2/3 when β≠0, under the condition αγ>0. These results extend global dynamics to low-regularity data for the coupled dispersive system, with implications for capillary-gravity and plasma-wave interactions.

Abstract

The Cauchy problem for a coupled system of the Schroedinger and the KdV equation is shown to be globally well-posed for data with infinite energy. The proof uses refined bilinear Strichartz estimates and the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao.

The Cauchy problem for a Schroedinger - Korteweg - de Vries system with rough data

TL;DR

This work establishes global well-posedness for a 1D Schrödinger–KdV system with rough initial data by combining refined bilinear Strichartz estimates with the I-method to create almost conserved energy functionals. The authors first secure local well-posedness in H^s for s>0 and then control the growth of modified energies E(Iu,Iv) and L(Iu,Iv) via detailed multilinear estimates and dyadic decompositions. By iterating short-time solutions and tuning the smoothing parameter N, they derive global results: s>3/5 when β=0 and s>2/3 when β≠0, under the condition αγ>0. These results extend global dynamics to low-regularity data for the coupled dispersive system, with implications for capillary-gravity and plasma-wave interactions.

Abstract

The Cauchy problem for a coupled system of the Schroedinger and the KdV equation is shown to be globally well-posed for data with infinite energy. The proof uses refined bilinear Strichartz estimates and the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao.

Paper Structure

This paper contains 7 sections, 10 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Bilinear Strichartz type estimates
  3. Local well-posedness
  4. Conserved and almost conserved quantities
  5. Estimates for the modified energy functional
  6. Estimates for the modified L - functional
  7. The global existence result

Key Result

Lemma 0.1

For 1/2 > b > b' \ge 0 , 0 < \delta \le 1 , m \in {\bf R} the following estimate holds:

Theorems & Definitions (10)

  • Lemma 0.1
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Theorem 2.1
  • Proposition 2.1
  • Theorem 3.1
  • Proposition 4.1
  • Proposition 5.1
  • Theorem 6.1