# Journal of the European Mathematical Society

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**Volume 17, Issue 1, 2015, pp. 71–149**

**DOI: 10.4171/JEMS/499**

Published online: 2015-02-05

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia^{[1]}and Vadim Kaloshin

^{[2]}(1) Universitat Politècnica de Catalunya, Barcelona, Spain

(2) University of Maryland, College Park, United States

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