No abstract

We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n 3. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the potentials, while the potentials themselves can be large. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials.

We prove almost optimal local well-posedness for the coupled Dirac-Klein-Gordon (DKG) system of equations in 1 + 3 dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman-Machedon type. It has been known for some time that the Klein-Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.

Abstract. We prove global smoothing and Strichartz estimates for the Schrö-dinger, wave, Klein-Gordon equations and for the massless and massive Dirac systems, perturbed with singular electromagnetic potentials. We impose a smallness condition on the magnetic part, while the electric part can be large. The decay and regularity assumptions on the coefficients are close to critical.

We uncover the full null structure of the Maxwell-Dirac system in Lorenz gauge. This structure, which cannot be seen in the individual component equations, but only when considering the system as a whole, is expressed in terms of tri-and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies. In the 3D case, we prove frequencylocalized L 2 space-time estimates for these integral forms at the scale invariant regularity up to a logarithmic loss, hence we obtain almost optimal local wellposedness of the system by iteration.