Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity (i∂t + ∆)u = ±∂(u m) on R d , d ≥ 1, with random initial data, where ∂ is a first order derivative with respect to the spatial variable, for example a linear combination of ∂ ∂x 1 ,. .. , ∂ ∂x d or |∇| = F −1 [|ξ|F ]. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in H s (R d) with s > max d−1 d sc, sc 2 , sc − d 2(d+1) for d + m ≥ 5, where s is below the scaling critical regularity sc := d 2 − 1 m−1 .

The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities

“…Recently, there are many researches, using probabilistic tools to study nonlinear dispersive equations in scaling supercritical regimes, for example [2,3,4,5,6,7,10,15,16,21]. Bényi-Oh-Pocovnicu [2] and Lührmann-Mendelson [21] introduced randomization on R d , independently.…”

“…Thus, it is possible to establish supercritical well-posedness of (1.1) with suitable random initial data. Hirayama and Okamoto [15] studied random data Cauchy problem for 4NLS (i∂ t + ∆ 2 )u = ±∂(|u| 2 u) below the scaling critical regualrity. Via Strichartz estimates and bilinear estimates, Chen and Zhang [7] investigated the random data Cauchy problem of 4NLS with nonlinearities containing the second order derivatives in H s (R d ).…”

Random data theory for the cubic fourth-order nonlinear Schrödinger equation

“…There are several works on random Cauchy theory followed these results (see e.g. [19,2,30,4,24,15]).…”

Almost Sure Global Well-posedness for the Fourth-order nonlinear Schrodinger Equation with Large Initial Data

“…Recently, there are many researchers, using probabilistic tools to study nonlinear dispersive equations in scaling supercritical regimes, for example [2,6,8,16,20,28,32]. This idea was brought by Bourgain [4,5] for the periodic nonlinear Schrödinger(NLS) equation.…”