Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics

Herr, Sebastian

Journées équations aux dérivées partielles (2010), article no. 10, 10 p.

This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds $M$ with small initial data in $H^1\left(M\right)$. The results include small data GWP for the quintic NLS in the case of the $3d$ flat rational torus $M={𝕋}^3$ and small data GWP for the corresponding cubic NLS in the cases $M={ℝ}^2\times {𝕋}^2$ and $M={ℝ}^3\times 𝕋$. The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling, as well as critical function space theory. All results mentioned above have been obtained in collaboration with D. Tataru and N. Tzvetkov.

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