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We obtain almost-sure scattering for the Schrödinger equation with a defocusing cubic nonlinearity in the Euclidean space \mathbb {R}^3, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [Trans. Amer. Math. Soc. Ser. B 2 (2015), pp. 1–50]. It also extends the results of Dodson, Lührmann and Mendelson [Adv. Math. 347 (2019), pp. 619–676] on the energy-critical equation in \mathbb {R}^4, to the energy-subcritical equation in \mathbb {R}^3. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect which makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao. Then, we propose a globalization argument in which we set up the I-method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime. [ABSTRACT FROM AUTHOR]

We investigate a regularity for weak solutions of the following generalized Leray equations (−Δ)α V − 2α−1/2α V + V ⋅ ∇ V − 1/2α x ⋅ ∇V + ∇P = 0, which arises from the study of self-similar solutions to the generalized Navier-Stokes equations in R3. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions V in the weighted Hilbert space Hαω(R3). Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from Hαω(R3) to Hω1+α(R3). This regularity result, together with linear theory for the non-local Stokes system, leads to pointwise estimates of V which allow us to obtain a natural pointwise property of the self-similar solution constructed by Lai, Miao, and Zheng [Adv. Math. 352 (2019), pp. 981–1043]. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Navier-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai Comm. Math. Phys., 328 (2014), pp. 29–44. [ABSTRACT FROM AUTHOR]