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

TitleGlobal, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials
Publication TypePreprint
2017
AuthorsAntonelli, P, Michelangeli, A, Scandone, R
Document NumberSISSA;17/2017/MATE

We prove the existence of weak solutions in the space of energy for
a class of non-linear Schördinger 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 suffcient
compactness properties in order to produce a global-in-time finite energy weak
solution to our original problem.

http://preprints.sissa.it/handle/1963/35294

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