Global well-posedness for the non-linear Maxwell-Schrödinger system

• Autores: Paolo Antonelli, Pierangelo Marcati, Raffaele Scandone
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 23, Nº 3, 2022, págs. 1293-1324
• Idioma: inglés
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• Resumen

  • In this paper we study the Cauchy problem associated to the Maxwell- Schr ̈odinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non- relativistic quantum plasma, interacting with its self-generated electromagnetic potential. One consequence of our analysis is to demonstrate that the Lorentz force associated with the electromagnetic field is well-defined for solutions slightly more regular than the finite energy class. This aspect is of fundamen- tal importance since all the related physical models require the observability of electromagnetic effects. The well-posedness of the Lorentz force still seems to be a major open problem in the class of solutions which are only finite energy.

    We show the global well-posedness at high regularity for the cubic and sub-cubic case, and we provide polynomial bounds for the growth of the Sobolev norm of the solutions, for a certain range of non-linearities. An important role is played by appropriate a priori dispersive estimates, obtained by means of Koch-Tzvetkov type bounds for the non-homogeneous Schr ̈odinger equation, which overcome the lack of Strichartz estimates for the magnetic Schr ̈odinger flow. Because of the power-type non-linearity, the propagation of higher regularity, globally in time, cannot be achieved via a bootstrap argument as done in [45]. Our approach then exploits the analysis of a modified energy functional, combined with the a priori bounds coming from the dispersive estimates obtained previously.