On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3

By 'Arp'ad B'enyi, Tadahiro Oh, Oana Pocovnicu. 2014

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We consider the Cauchy problem of the cubic nonlinear Schrodinger equation (NLS) : i∂tu+Δu = ±|u|2u on Rd, d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity scrit = d−2 2 . More precisely, given a function on Rd, 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4afff8c4e70279d5590fb54007d43e291c8e6f4e