Global well-posedness for the massless cubic Dirac equation

Abstract

We show that the cubic Dirac equation with zero mass is globally well-posed for small data in the scale invariant space H^{\frac{n-1}{2}}(R^n) for n=2, 3. The proof proceeds by using the Fierz identities to rewrite the equation in a form where the null structure of the system is readily apparent. This null structure is then exploited via bilinear estimates in spaces based on the null frame spaces of Tataru. We hope that the spaces and estimates used here can be applied to other nonlinear Dirac equations in the scale invariant setting. Our work complements recent results of Bejenaru-Herr who proved a similar result for n=3 in the massive case.

Error in the statement and proof of Theorem 4.1 has been corrected

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Cited by in corpus (4)

• The cubic Dirac equation: Small initial data in $H^{\frac{1}{2}}({\mathbb{R}}^2)$
• The cubic Dirac equation: Small initial data in $H^1({\mathbb{R}}^3)$
• On the Division Problem for the Wave Maps Equation
• On the Majorana condition for nonlinear Dirac systems