The cubic Dirac equation: Small initial data in $H^1({\mathbb{R}}^3)$

Abstract

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space $H^1({\mathbb{R}}^3)$. 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.

v3: minor revision; 47 pages

References in corpus (4)

• Energy dispersed large data wave maps in 2+1 dimensions
• The cubic Dirac equation: Small initial data in $H^{\frac{1}{2}}({\mathbb{R}}^2)$
• Global well-posedness for the massless cubic Dirac equation
• A counterexample to an endpoint bilinear Strichartz inequality

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• On the one dimensional Dirac equation with potential
• On the Majorana condition for nonlinear Dirac systems
• Small data scattering of 2d Hartree type Dirac equations
• Local smooth solutions of the nonlinear Klein-gordon equation
• The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions
• Local well-posedness of Dirac equations with nonlinearity derived from honeycomb structure in 2 dimensions
• Global well-posedness of high dimensional Maxwell-Dirac for small critical data
• Decay of solutions of nonlinear Dirac equations
• Ill-posedness of the Thirring model below the critical regularity