Journal of the European Mathematical Society


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Volume 17, Issue 1, 2015, pp. 71–149
DOI: 10.4171/JEMS/499

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia[1] and Vadim Kaloshin[2]

(1) Departament de Matemàtiques, Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028, Barcelona, Spain
(2) Department of Mathematics, University of Maryland, 3111 Mathematics Building, MD 20740, College Park, USA

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

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

Keywords: Hamiltonian partial differential equations, nonlinear Schrödinger equation, transfer of energy, growth of Sobolev norms, normal forms of Hamiltonian fixed points

Guardia Marcel, Kaloshin Vadim: Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation. J. Eur. Math. Soc. 17 (2015), 71-149. doi: 10.4171/JEMS/499