We study the global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation (NLS), i∂ t u + ∆u + |u| p−1 u = 0, with initial data u 0 ∈ H 1 , x ∈ R d . The nonlinearity power p and the dimension d are such that the scaling index s = d 2 − 2 p−1 is between 0 and 1, thus, the NLS is mass-supercritical (s > 0) and energy-subcritical (s < 1).For solutions with ME[u 0 ] < 1 (ME[u 0 ] stands for an invariant and conserved quantity in terms of the mass and energy of u 0 ), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient G u of a solution u to NLS is initially less than 1, i.e., G u (0) < 1, then the solution exists globally in time and scatters in H 1 (approaches some linear Schrödinger evolution as t → ±∞); if the renormalized gradient G u (0) > 1, then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of H 1 norm in infinite time.This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle.

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We rule out the existence of Leray's backward self-similar solutions to the Navier-Stokes equations with profiles in L 12/5 (R 3 ) or in the Marcinkiewicz space L q,∞ (R 3 ) for q ∈ (12/5, 6). This follows from a more general result formulated in terms of Morrey spaces and the first order Riesz's potential.

Abstract. Using the concentration-compactness method and the localized virial type arguments, we study the behavior of H 1 solutions to the focusing quintic NLS in R 2 , namely, Denoting by M [u] and E[u], the mass and energy of a solution u, respectively, and Q the ground state solution to −Q+∆Q+|Q| 4 Q = 0, and assuming