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Asymptotic decay for the Chern-Simons-Higgs equations

Dongyi Wei, Shiwu Yang

Abstract

In this paper, we study the long time asymptotic behaviors for solutions to the Chern-Simons-Higgs equation with a pure power defocusing nonlinearity. We obtain quantitative inverse polynomial time decay for the potential energy for all data with finite conformal energy. Consequently, the solution decays in time in the pointwise sense for all power. We also show that for sufficiently large power the solution decays as quickly as linear waves. Key ingredients for the proof include vector field method, conformal compactification and the geometric bilinear trace theorem for null hypersurface developed by Klainerman-Rodnianski.

Asymptotic decay for the Chern-Simons-Higgs equations

Abstract

In this paper, we study the long time asymptotic behaviors for solutions to the Chern-Simons-Higgs equation with a pure power defocusing nonlinearity. We obtain quantitative inverse polynomial time decay for the potential energy for all data with finite conformal energy. Consequently, the solution decays in time in the pointwise sense for all power. We also show that for sufficiently large power the solution decays as quickly as linear waves. Key ingredients for the proof include vector field method, conformal compactification and the geometric bilinear trace theorem for null hypersurface developed by Klainerman-Rodnianski.
Paper Structure (8 sections, 16 theorems, 291 equations)

This paper contains 8 sections, 16 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Potential energy decay
  3. Rough pointwise decay estimate for all $p$
  4. Improved pointwise decay estimates for larger $p$
  5. Conformal transformation
  6. Singular Chern-Simons-Higgs equation on compact region
  7. Geometric Kirchoff-Sobolev paramatrix and sharp decay estimates
  8. Proof for the main theorem \ref{['thm:main:pd:imp']}

Key Result

Theorem 1

Consider the Cauchy problem to the Chern-Simons-Higgs equation eq:CSH:p:2d in $\mathbb{R}^{1+2}$ with a pure power defocusing nonlinearity and finite initial conformal energy $\mathcal{E}_{0, 2}$. Then the potential energy decays inverse polynomially in time for some constant $C$ depending only on $p$.

Theorems & Definitions (30)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Theorem 3
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Remark 1
  • Lemma 3.1
  • Lemma 3.2
  • ...and 20 more