The Perturbative Approach to the Nonlinear Problem

Theorem 1.1. Consider (NLS) on Rd or Td, with μ = ±1. Fix s > d2 ; then (NLS) is locally well-posed in H s (in the sense of mild solutions): • Existence: For any u0 ∈ Hs, there exists a (mild) solution u defined on [0, T0], where T0 = cs‖u0‖ Hs, for a constant cs > 0. It is such that ‖u‖CT0Hs ≤ 2‖u0‖Hs. • Uniqueness: it is unique in LT0H s. • Dependence on the data: the map u0 7→ u is analytic from Hs to CT0H (more precisely: the time of existence T0 can be chosen locally uniformly, and around any u0 the solution operator can be expanded in series of multilinear operators).