The Cubic Dirac Equation: Small Initial Data in H¹(ℝ³) H 1 ( R 3 )

2013/10/31 by Ioan Bejenaru, Sebastian Herr · 3 citations
Mathematics · #Advanced Mathematical Physics Problems #Mathematical Analysis and Transform Methods #math.AP #msc:35Q41

paper · pdf · doi:10.1007/s00220-014-2164-0

published as Comm. Math. Phys. (2015), Vol. 335, No. 1, 43-82 · v3: minor revision; 47 pages

arxiv created 2014/08/06 · openalex publication_date 2014/09/18 · arxiv updated 2015/03/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space H¹(ℝ³). The main ingredient is obtaining a sharp end-point Strichartz estimate for the Klein-Gordon equation. In a classical sense this fails and it is related to the failure of the endpoint Strichartz estimate for the wave equation in space dimension three. In this paper, systems of coordinate frames are constructed in which endpoint Strichartz estimates are recovered and energy estimates are established.

Citations

• Global well-posedness for the massless cubic Dirac equation
• Well-posedness and scattering for the KP-II equation in a critical space
• Normal forms and quadratic nonlinear Klein‐Gordon equations

Cited by

• Transference of bilinear restriction estimates toquadratic variation norms and the Dirac–Klein–Gordon system
• The Cubic Dirac Equation: Small Initial Data in H^(1)/(2) (ℝ²)
• On the Division Problem for the Wave Maps Equation

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• The cubic nonlinear Dirac equation
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• Scattering results for Dirac Hartree-type equations with small initial data