Growth of Sobolev norms for 2d NLS with harmonic potential

• Fabrice Planchon ^[1] ; Nikolay Tzvetkov ^[3] ; Nicola Visciglia ^[2]
  1. [1] Pierre and Marie Curie University

     Pierre and Marie Curie University

     París, Francia

     
  2. [2] University of Pisa

     University of Pisa

     Pisa, Italia

     
  3. [3] ENS Lyon

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 4, 2023, págs. 1405-1436
• Idioma: inglés
• DOI: 10.4171/RMI/1371
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• Resumen

  • We prove polynomial upper bounds on the growth of solutions to the 2d cubic nonlinear Schrödinger equation where the Laplacian is confined by the harmonic potential. Due to better bilinear effects, our bounds improve on those available for the 2d cubic nonlinear Schrödinger equation in the periodic setting: our growth rate for a Sobolev norm of order s is t2(s−1)/3+ε, for s=2k and k≥1 integer. In the appendix we provide a direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.