Global well-posedness and scattering for the defocusing quintic NLS in three dimensions

We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments.This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.By Sobolev embedding, u(0) has finite energy if and only if u(0) ∈ Ḣ 1x ‫ޒ(‬ 3 ), which is the space of initial data that we consider.This is also a scale-invariant space; both the class of solutions to (1-1) and the energy are invariant under the scaling symmetry u(t, x) → u λ (t, x) := λ 1/2 u(λ 2 t, λx).(1-3) For this reason, the equation is termed energy-critical.A function u : I × ‫ޒ‬ 3 → ‫ރ‬ on a nonempty time interval I 0 is called a strong solution to (1-1) if it lies in the class) for all compact K ⊂ I , and obeys the Duhamel formulafor all t ∈ I .We say that u is a maximal-lifespan solution if the solution cannot be extended (in this class) to any strictly larger interval.Our main result is a new proof of the following: