Refined probabilistic local well-posedness for a cubic Schrödinger half-wave equation

We obtain probabilistic local well-posedness in quasilinear regimes for the Schrodinger half-wave equation with a cubic nonlinearity. We need to use a refined ansatz because of the lack of probabilistic smoothing in the Picard's iterations, which is due to the high-low-low nonlinear interactions. The proof is an adaptation of the method of Bringmann on the derivative nonlinear wave equation [6] to Schrodinger-type equations. In addition, we discuss ill-posedness results for this equation. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we consider the probabilistic local well-posedness problem for the Schrodinger half-wave equation with a cubic nonlinearity in quasilinear regimes. Due to the lack of probabilistic smoothing in the Picard's iterations caused by high-low-low nonlinear interactions, we need to use a refined ansatz. The proof is an adaptation of Bringmann's method on the derivative nonlinear wave equation [6] to Schrodinger-type equations. In addition, ill-posedness results for this equation are discussed.