Global solution to the cubic Dirac equation in two space dimensions

Abstract

We are interested in the cubic Dirac equation with mass m∈[0,1] in two space dimensions, which is also known as the Soler model. We conduct a thorough study on this model with initial data sufficiently small in high regularity Sobolev spaces. First, we show the global existence of the cubic Dirac equation, which is uniform-in-mass in the sense that the smallness condition on the initial data is independent of the mass parameter m. In addition, we derive a unified pointwise decay result valid for all m∈[0,1]. Last but not least, we prove solution to the cubic Dirac equation scatters linearly. When the mass m=0, we can show an improved pointwise decay result.