The initial value problem for Schrödinger equations on the torus

where u(t, x) is a complex valued unknown function of (t, x) = (t, x1, . . . , xn) ∈ R × T, T = R/2πZ, i = √−1, ∂t = ∂/∂t, ∂j = ∂/∂xj (j = 1, . . . , n), ∇ = (∂1, · · · , ∂n), ∆ = ∇ · ∇, and ~b(x) = (b1(x), . . . , bn(x)), c(x), f(t, x) and u0(x) are given functions. Suppose that b1(x), . . . , bn(x) and c(x) are smooth functions on T , and that F (u, v, ū, v̄) is a smooth function on R, and F (u, v, ū, v̄) = O(|u|2 + |v|2) near (u, v) = 0. For the Euclidean case x ∈ R, Mizohata proved in [8] that if the initial value problem (1)-(2) is L-well-posed, then