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

  • Paolo Antonelli
  • Alessandro Michelangeli
  • Raffaele Scandone
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Abstract

We prove the existence of weak solutions in the space of energy for a class of nonlinear Schrödinger equations in the presence of a external, rough, time-dependent 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 regularisation, 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.

Keywords

Nonlinear Schrödinger equation Magnetic potentials Viscosity regularisation Strichartz estimates Weak solutions 

Mathematics Subject Classification

35D40 35H30 35Q41 35Q55 35K08 
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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Paolo Antonelli
    • 1
  • Alessandro Michelangeli
    • 2
  • Raffaele Scandone
    • 2
  1. 1.Gran Sasso Science Institute – GSSIL’AquilaItaly
  2. 2.SISSA – International School for Advanced StudiesTriesteItaly

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