Endpoint Strichartz estimates for the magnetic Schrödinger equation

Strichartz Estimates for the Magnetic Schrödinger Equation

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n ≥ 6.

Global Strichartz estimates for approximations of the Schrödinger equation

We consider a semidiscrete scheme for the linear Schrödinger equation with high order dissipative term. We obtain maximum norm estimates for its solutions and we prove global Strichartz estimates for the considered model, estimates that are uniform with respect to the mesh size. The methods we employ are based on classical arguments of harmonic analysis.

Counterexamples to Strichartz Estimates for the Magnetic Schrödinger Equation

In space dimension n ≥ 3, we consider the magnetic Schrödinger Hamiltonian H = −(∇− iA(x)) and the corresponding Schrödinger equation i∂tu + Hu = 0. We show some explicit examples of potentials A, with less than Coulomb decay, for which any solution of this equation cannot satisfy Strichartz estimates, in the whole range of Schrödinger admissibility.

Strichartz Estimates for the Schrödinger Equation

The objective of this paper is to report on recent progress on Strichartz estimates for the Schrödinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in Wiener amalgam and modulation spaces. We present and compare the different technicalities. Then, we illustrate applications to well-posedness.

Characteristic Strichartz Estimates for the Wave Equation