Blow-up solutions on a sphere for the 3d quintic NLS in the energy space

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We prove that if u(t) is a log-log blow-up solution, of the type studied by Merle-Raphaël (2001-2005), to the L² critical focusing NLS equation i∂ₜ u +Δu + |u|⁴/ᵈ u=0 with initial data u₀∈ H¹(Rᵈ) in the cases d=1, 2, then u(t) remains bounded in H¹ away from the blow-up point. This is obtained without assuming that the initial data u₀ has any regularity beyond H¹(Rᵈ). As an application of the d=1 result, we construct an open subset of initial data in the radial energy space H¹rad(R³) with corresponding solutions that blow-up on a sphere at positive radius for the 3d quintic (H¹-critical) focusing NLS equation i∂ₜu + Δu + |u|⁴u=0. This improves Raphaël-Szeftel (2009), where an open subset in H³rad(R³) is obtained. The method of proof can be summarized as follows: on the whole space, high frequencies above the blow-up scale are controlled by the bilinear Strichartz estimates. On the other hand, outside the blow-up core, low frequencies are controlled by finite speed of propagation.