Lecture notes : Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations

On the global well-posedness of focusing energy-critical inhomogeneous NLS

“…The condition (1.10) leads us to the inequality (|∇u| 2 −(1−η)g|u| p+1 ) dx < −C for some 0 ≤ η < 1 and C > 0 and hence to the finite time blowup through the localized virial identity (6.5) below. This argument also appears in some literatures (see [21,2,13]). In (2) the moment condition |x|ϕ ∈ L 2…”

Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation

“…We are indebted to M. Visan and X. Zhang and to J. Holmer and S. Roudenko, for pointing this out to us. A correct proof is given in [18].…”

“…(Long time perturbation theory; see also [18], [19] and [45]) Let I ⊂R be a time interval. Let t 0 ∈I, (u 0 , u 1 )∈Ḣ 1 ×L 2 and some constants M, A, A >0 be given.…”

The Concentration-Compactness Rigidity Method for Critical Dispersive and Wave Equations

“…This says that, for t ∈ I, ∇u(t) · ∂ t u(t) = ∇u 0 · u 1 . Finally, the analog of the "Perturbation Theorem" also holds in this context (see [22]). All the corollaries of the Perturbation Theorem also hold.…”