We study coupled systems of nonlinear lowest Landau level equations, for which we prove global existence results with polynomial bounds on the possible growth Sobolev norms solutions. also exhibit explicit unbounded trajectories show that these are optimal.

We prove polynomial upper bounds on the growth of solutions to $2d$ cubic nonlinear Schrödinger equation where Laplacian is confined by harmonic potential. Due better bilinear effects, our improve those available for in periodic setting: rate a Sobolev norm order $s$ $t^{2(s-1)/3+\varepsilon}$, $s=2k$ and $k\geq 1$ integer. In appendix we provide direct proof, based integration parts, estimates...