Advanced Search

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...