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Abstract
We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) : i∂_t u+Δu = ±u2u on ℝ^d, d ≥ 3, with random initial data and prove almost sure wellposedness results below the scalingcritical regularity s_{crit} = (d−2)/2. More precisely, given a function on ℝ^d, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the socalled modulation spaces. Our goal in this paper is threefold. (i) We prove almost sure local wellposedness of the cubic NLS below the scaling critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove ‘conditional’ almost sure global wellposedness 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 wellposedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global wellposedness and scattering with a large probability for initial data randomized on dilated cubes.
Original language  English 

Pages (fromto)  150 
Number of pages  50 
Journal  Transactions of the American Mathematical Society: Series B 
Volume  2 
DOIs  
Publication status  Published  26 May 2015 
Keywords
 nonlinear Schrödinger equation
 almost sure wellposedness
 modulation space
 Wiener decomposition
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Dive into the research topics of 'On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^d, d ≥ 3'. Together they form a unique fingerprint.Projects
 1 Finished

ProbDynDispEq  Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20
Project: Research