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Scattering for the magnetic Zakharov system in 3 dimensions

Xiaohong Wang, Lijia Han

Abstract

We consider the global existence and scattering for solutions of magnetic Zakharov system in three-dimensional space. When the initial data is small, we prove the existence of smooth global solutions and scattering results, by combining the space--time resonance method, weighted Sobolev space and dispersive estimates. Moreover, the decay rates for the solutions are also obtained.

Scattering for the magnetic Zakharov system in 3 dimensions

Abstract

We consider the global existence and scattering for solutions of magnetic Zakharov system in three-dimensional space. When the initial data is small, we prove the existence of smooth global solutions and scattering results, by combining the space--time resonance method, weighted Sobolev space and dispersive estimates. Moreover, the decay rates for the solutions are also obtained.
Paper Structure (44 sections, 28 theorems, 387 equations)

This paper contains 44 sections, 28 theorems, 387 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Resonance analysis and resolution space
  5. Space-time resonance set
  6. Resolution space
  7. Energy Estimates
  8. Weighted Estimates
  9. Decay Estimate
  10. Weighted Estimates for the Schr$\ddot{o}$dinger Component
  11. Introduction
  12. Main achievements
  13. Sketch of the proof
  14. Preliminaries
  15. Notations
  16. ...and 29 more sections

Key Result

Theorem 1.1

When $\beta\neq1$, suppose that the initial datas $\mathcal{E}_{0}$, $n_{0}$, $n_{1}$, $\mathcal{B}_{0}$, $\mathcal{B}_{1}$ satisfy for some small $\epsilon_{0}$ and some large integer $N$. Then the Cauchy problem for the magnetic Zakharov system 1.1 admits a unique global solution such that where the parameters could be chosen as Furthermore, let $(\mathcal{E},n,\mathcal{B})(t)$ are the soluti

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Proposition 4.1
  • ...and 21 more