An instability property of the nonlinear Schrödinger equation on $S^{d}$

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Nonlinear wave interactions for the Benjamin-Ono equation. (2004, methodology): “In particular we use only the first few integrals of the Benjamin-Ono and we do this only for the range 0 s ≤ 3/2.”
Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $S^1$ (2010, methodology): “A similar construction was used in [21] to prove instability properties in Sobolev spaces of negative index.”
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations (2002, methodology): “The first of these three conclusions has already been obtained by Burq, Gérard and Tzvetkov [8].”
Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr (2004, background): “The approach on R3 breaks down in the case of a compact manifold since the corresponding Strichartz type estimates have to encounter some unavoidable derivative losses (see [1, 5, 6]).”