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

We prove smoothing estimates in Morrey-Campanato spaces for a Helmholtz equationwith fully variable coefficients, of limited regularity, defined on the exterior of a starshaped compact obstacle in R n , n ≥ 3, with Dirichlet boundary conditions. The principal part of the operator is a long range perturbation of a constant coefficient operator, while the lower order terms have an almost critical decay. We give explicit conditions on the size of the perturbation which prevent trapping.As an application, we prove smoothing estimates for the Schrödinger flow e itL and the wave flow e it √ L with variable coefficients on exterior domains and Dirichlet boundary conditions.

We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.

In this paper, we build a Gibbs measure for the cubic defocusing Schrödinger equation on the real line with a decreasing interaction potential, in the sense that the non linearity |u| 2 u is multiplied by a function χ which we assume integrable and smooth enough. We prove that this equation is globally well-posed in the support of this measure and that the measure is invariant under the flow of the equation. What is more, the support of the measure (the set of initial data) is disjoint from L 2 .

Abstract. We prove local smoothing estimates for the massless Dirac equation with a Coulomb potential in 2 and 3 dimensions. Our strategy is inspired by [9] and relies on partial wave subspaces decomposition and spectral analysis of the Dirac-Coulomb operator.