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Small energy scattering for radial solutions to the generalized Zakharov system

Jun Kato, Osamu Tojo

Abstract

We prove the small energy scattering for the three-dimensional generalized Zakharov system with radial symmetry based on the idea by Guo and Nakanishi (2014), which treats the usual Zakharov system. For the proof, we use the frequency-localized normal form reduction, and the radially improved Strichartz estimates. The relation between the solution to the integral equations, which includes the unusual boundary terms, and the original differential equations is also considered.

Small energy scattering for radial solutions to the generalized Zakharov system

Abstract

We prove the small energy scattering for the three-dimensional generalized Zakharov system with radial symmetry based on the idea by Guo and Nakanishi (2014), which treats the usual Zakharov system. For the proof, we use the frequency-localized normal form reduction, and the radially improved Strichartz estimates. The relation between the solution to the integral equations, which includes the unusual boundary terms, and the original differential equations is also considered.
Paper Structure (16 sections, 12 theorems, 178 equations)

This paper contains 16 sections, 12 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Reduction of the system
  3. Notation
  4. The Schrödinger part
  5. The wave part
  6. Main Results
  7. Basic estimates
  8. Radial Strichartz estimates
  9. Bilinear Fourier multiplier
  10. Estimates of the nonlinear terms
  11. Quadratic terms
  12. Boundary terms
  13. Cubic terms
  14. Proof of Theorem \ref{['thm-1']}
  15. Proof of Theorem \ref{['thm-2']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\gamma\in [\frac{1}{2},1]$. We assume that $u_{0}$, $N_{0}$ are radial and satisfy $\lVert u_{0}\rVert_{H^{1}}+\lVert N_{0}\rVert_{H^{\frac{1-\gamma}{2}}}\le \rho$. If $\rho>0$ is sufficiently small, then there exists a unique global solution to int-normal-schrodinger, int-normal-wave satisfying Moreover, there exists $u_{+}\in H^{1}$ and $N_{+}\in H^{\frac{1-\gamma}{2}}$ such that

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Proposition 2.1
  • Remark 2.1
  • Proposition 2.2: guo-nakanishi_1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 15 more