Endpoint estimates and global existence for the nonlinear Dirac equation with potential

Abstract

We prove endpoint estimates with angular regularity for the wave and Dirac equations perturbed with a small potential. The estimates are applied to prove global existence for the cubic Dirac equation perturbed with a small potential, for small initial H-1 data with additional angular regularity. This implies in particular global existence in the critical energy space H-1 for small radial data. (c) 2012 Elsevier Inc. All rights reserved

Local smoothing estimates for the massless Dirac–Coulomb equation in 2 and 3 dimensions

“…[8], [13]). The angular endpoint estimates proved in [31] have been generalized in [11], [12] to small potential perturbations, by introducing some new mixed Strichartz-smoothing estimates. At the same time, some effort has been spent in order to find examples of potentials such that the correspoding flows do not disperse (in the sense discussed above).…”

On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential

“…Furthermore we have η = η + A(t, x) such that, for preassigned p 0 > 2, for all p ≥ p 0 and for 2 p = 3 2 1 − 2 q , (1. 12) we have z L ∞ t (R+) + η (1.13) (for the Besov spaces B k p,q see Sect. 2) and such that A(t, ·) ∈ Σ 4 for all t ≥ 0, with lim t→+∞ A(t, ·) Σ4 = 0.…”

On the cubic Dirac equation with potential and the Lochak–Majorana condition

“…The first estimate is precisely (2.36) of Corollary 2.4 in [7]. In order to prove (3.4), we argue exactly as in the proofs of Theorem 2.3 and Corollary 2.4 in [7], expanding the flow in spherical harmonics. The only modification is to replace the estimate after formula (2.30) in that paper with the following one:…”

“…The first goal of the paper is to study the dispersive properties of the Dirac flow perturbed by a large potential, and to prove several smoothing and (endpoint) Strichartz estimates for it. We then apply the estimates to prove the global existence of small solutions for the nonlinear equation (1.1), for H 1 initial data with additional angular regularity, in the spirit of [18] and [7]. Moreover, if the potential has an additional structure, we are able to reduce the smallness assumption to smallness of the chiral component of the initial data; to this end we exploit the Lochak-Majorana condition.…”