Abstract. We consider the Cauchy problem of the cubic nonlinear Schrö-dinger equation (NLS) : i∂ t u + Δu = ±|u| 2 u on R d , d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity. More precisely, given a function on R d , we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scalingcritical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove 'conditional' almost sure global well-posedness for d = 4 in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when d = 4, we show that conditional almost sure global wellposedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol e i|ξ | α , where α ∈ [0, 2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual L p -spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers |ξ | −δ sin(|ξ | α ) for 0 δ α.

We consider a randomization of a function on R d that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energycritical cubic nonlinear Schrödinger equation on R 4 is almost surely locally well-posed with respect to randomized initial data below the energy space.2010 Mathematics Subject Classification. 42B35, 35Q55.

We consider the cubic nonlinear Schrödinger equation (NLS) on R 3 with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.

By using tools of time‐frequency analysis, we obtain some improved local well‐posedness results for the nonlinear Schrödinger, nonlinear wave and nonlinear Klein–Gordon equations with Cauchy data in modulation spaces ℳ0,sp,1.

No abstract

Abstract. Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear Calderón-Zygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz' rule are presented as well.

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.2010 Mathematics Subject Classification. 35Q55, 35L71, 60H30.