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We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is such that for any we find a solution and a time such that . Moreover, the time satisfies the polynomial bound .
Guardia, 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 -
