Well-posedness for the cubic NLS on tori

The cubic nonlinear Schrodinger equation (NLS) is energy-critical (s_c = 1) with respect to the scaling symmetry, where s_c is the scaling critical regularity. The initial value problem (IVP) of cubic NLS is scaling invariant in the Sobolev norm H^1 of scaling critical regularity. First this talk introduce the deterministic global well-posedness result of cubic NLS on 4d-torus (T^4) in the critical regime (with H^1 initial data). Second we consider the cubic NLS in the super-critical regime (with H^s data, s1). A probabilistic approach is applied to obtain an “almost sure” well-posedness result for the cubic NLS on tori (T^d, d>=3).