# Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia; Vadim Kaloshin

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 1, page 71-149
- ISSN: 1435-9855

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topGuardia, Marcel, and Kaloshin, Vadim. "Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation." Journal of the European Mathematical Society 017.1 (2015): 71-149. <http://eudml.org/doc/277324>.

@article{Guardia2015,

abstract = {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 $\Vert u(T)\Vert _\{H^s\}\ge \mathcal \{K\} \Vert u(0)\Vert _\{H^s\}$. Moreover, the time $T$ satisfies the polynomial bound $0<T<\mathcal \{K\}^C$.},

author = {Guardia, Marcel, Kaloshin, Vadim},

journal = {Journal of the European Mathematical Society},

keywords = {Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed points; Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed points},

language = {eng},

number = {1},

pages = {71-149},

publisher = {European Mathematical Society Publishing House},

title = {Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation},

url = {http://eudml.org/doc/277324},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Guardia, Marcel

AU - Kaloshin, Vadim

TI - Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 1

SP - 71

EP - 149

AB - 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 $\Vert u(T)\Vert _{H^s}\ge \mathcal {K} \Vert u(0)\Vert _{H^s}$. Moreover, the time $T$ satisfies the polynomial bound $0<T<\mathcal {K}^C$.

LA - eng

KW - Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed points; Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed points

UR - http://eudml.org/doc/277324

ER -

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