Global Well-Posedness for the Massless Cubic Dirac Equation: Table 1

Abstract

We show that the cubic Dirac equation with zero mass is globally well-posed for small data in the scale invariant space 9 H n´1 2 pR n q 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.

The Cubic Dirac Equation: Small Initial Data in $${{H^{\frac{1}{2}}} (\mathbb{R}^{2}}$$)

“…A special case arises in the massless variant of the cubic Dirac equation, that is (1.2) with M = 0. A recent result of Bournaveas and Candy [3] provides the equivalent result of Theorem 1.1 for the case M = 0. Their strategy stems from the observation that the massless case carries similarities to the Wave Maps equation.…”

“…As in dimension n = 3 in [1], combining the energy and the Strichartz estimate to derive a uniform L 2 estimate is only possible in the presence of a null structure, see Sect. 3.…”

“…In dimension n = 2 and M = 0, we are aware of only two results: [19,20] where Pecher establishes local well-posedness of the equation with initial data in H s (R 2 ) for s > 1 2 and [3] where Bournaveas and Candy establish local well-posedness of the equation with initial data in H 1 2 (R 2 ). To our best knowledge, no global well-posedness result is known so far.…”

“…To our best knowledge, no global well-posedness result is known so far. The case M = 0 has been settled in [3] where Bournaveas and Candy also prove global well-posedness and scattering for small initial data inḢ 1 2 (R 2 ), see more commentaries below about this case.…”

“…The authors tailor their resolution spaces around the original ones introduced by Tataru [26] in the context of Wave Maps. In order to overcome the Besov space obstacle, the authors of [3] used an idea from [1], i.e., adding a high modulation nonlinear structure of type L p t L 2…”