Remarks on the probabilistic well-posedness for quadratic nonlinear Schrodinger equations (Harmonic Analysis and Nonlinear Partial Differential Equations)

Abstract

We consider the Cauchy problem for the quadratic nonlinear Schrodinger equation without gauge invariance: ipartial_{t}u+Delta u= |u|^{2} . First, we show the probabilistic well‐posedness in H^{s}(mathbb{R}^{d}) for d geq 5 and frac{d-3}{d-2}s_{c} < s < s_{c} , where s_{c} := frac{d}{2} -2 is the scaling critical regularity. Second, as in the paper of Bényi et al., by performing a fixed point argument around the higher order expansion, we improve the regularity threshold for almost sure local well‐posedness, i.e., frac{d-3}{d-2}s_{c} is replaced by frac{d-4}{d-3}s_{c}.