Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials

TL;DR

This paper establishes the existence of global finite energy weak solutions for nonlinear Schrödinger equations with rough, time-dependent magnetic potentials, using parabolic regularization and smoothing estimates.

Contribution

It introduces a novel approach employing parabolic regularization and dissipative estimates to handle rough magnetic potentials where traditional methods fail.

Findings

Existence of global finite energy weak solutions.

Applicable to rough, time-dependent magnetic potentials.

Method bypasses resolvent and Fourier techniques.

Abstract

We prove the existence of weak solutions in the space of energy for a class of non-linear Schroedinger equations in the presence of a external rough magnetic potential. Under our assumptions it is not possible to study the problem by means of usual arguments like resolvent techniques or Fourier integral operators, for example. We use a parabolic regularization and we solve the approximating Cauchy problem. This is achieved by obtaining suitable smoothing estimates for the dissipative evolution. The total mass and energy bounds allow to extend the solution globally in time. We then infer sufficient compactness properties in order to produce a global-in-time finite energy weak solution to our original problem.