Bounds on Sobolev norms for the nonlinear Schr\"odinger equation on   general tori
  We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schr\"odinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove polynomial bounds on the possible growth of Sobolev norms of smooth solutions. 
F. Catoire and W.-M. Wang