We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on d-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in L 2 (T). As for other power-type nonlinearities, namely (i) (super)quintic when d = 1 and (ii) (super)cubic when d ≥ 2, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.Date: March 8, 2018. 2010 Mathematics Subject Classification. 60H15. Key words and phrases. stochastic nonlinear Schrödinger equations; well-posedness. 1. The multiplicative noise given by the Stratonovich product u • φξ with real-valued ξ is relevant in physical applications, as it conserves the mass of u (i.e. t → u(t) 2 L 2x (T d ) is constant) almost surely. Our analysis can handle either the Itô or the Stratonovich product, and we choose to work with the former for the sake of simpler exposition.Here, P ≤N is the Littlewood-Paley projection onto frequencies {n ∈ Z d : |n| ≤ N }, p ≥ 2(d+2) d , and ε > 0 is an arbitrarily small quantity 3 . However, such Strichartz estimates are not strong enough for a fixed point argument in mixed Lebesgue spaces for the deterministic NLS on T d . To overcome this problem, we shall employ the Fourier restriction norm method by means of X s,b -spaces defined via the norms(1.7)The indices s, b ∈ R measure the spatial and temporal regularities of functions u ∈ X s,b , and F t,x denotes Fourier transform of functions defined on R × T d . This harmonic analytic 2. Here, W s,r (T d ) denotes the L r -based Sobolev space defined by the Bessel potential norm:, where n := 1 + |n| 2 . When r = 2, we have H s (T d ) = W s,2 (T d ).3. More recently, Killip and Vişan [25] removed the arbitrarily small loss of ε derivatives in (1.6) when p > 2(d+2) d . However, we do not need this scale-invariant improvement in our results.

Presented by Haïm BrézisIn this note, we consider the derivative nonlinear Schrödinger equation on the circle. In particular, by adapting Wu's recent argument to the periodic setting, we prove its global well-posedness in H 1 (T), provided that the mass is less than 4π . Moreover, this mass threshold is independent of spatial periods.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m éOn considère dans cette note l'équation de Schrödinger avec dérivée sur le cercle. En particulier, en adaptant l'argument récent de Wu au cas periodique, on prouve que cette équation est globalement bien posée dans H 1 (T), pourvu que la masse soit inférieure à 4π . En outre, ce seuil pour la masse est indépendant des périodes spatiales.

No abstract

Abstract. We establish the global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in the Sobolev space H 1 2 , provided that the mass of initial data is less than 4π. This result matches the one by Miao, Wu, and Xu and its recent mass threshold improvement by Guo and Wu in the non-periodic setting. Below H 1 2 , we show that the uniform continuity of the solution map on bounded subsets of H s does not hold, for any gauge equivalent equation.