JournalsjemsVol. 17, No. 1pp. 71–149

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

  • Marcel Guardia

    Universitat Politècnica de Catalunya, Barcelona, Spain
  • Vadim Kaloshin

    University of Maryland, College Park, United States
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Abstract

We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix s>1s>1. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with ss-Sobolev norm growing in time.

We establish the existence of solutions with polynomial time estimates. More exactly, there is c>0c>0 such that for any K1\mathcal K\gg 1 we find a solution uu and a time TT such that u(T)HsKu(0)Hs\| u(T)\|_{H^s}\geq\mathcal K \| u(0)\|_{H^s}. Moreover, the time TT satisfies the polynomial bound 0<T<Kc0 < T < \mathcal K^c.

Cite this article

Marcel Guardia, Vadim Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation. J. Eur. Math. Soc. 17 (2015), no. 1, pp. 71–149

DOI 10.4171/JEMS/499