Almost sure local wellposedness of energy critical fractional Schr\"odinger equations with hartree nonlinearity

TL;DR

This paper proves that the energy-critical fractional Schrödinger equation with Hartree nonlinearity is almost surely locally well-posed below the energy space using a randomization method, extending results to the classical Hartree Schrödinger case.

Contribution

It introduces a probabilistic approach to establish local well-posedness for fractional Schrödinger equations below the energy space, including the classical Hartree case.

Findings

Almost sure local well-posedness is established.

Method applies to fractional Schrödinger equations with Hartree nonlinearity.

Includes the classical Hartree Schrödinger equation as a special case.

Abstract

We consider a Cauchy problem of energy-critical fractional Schr\"odinger equation with Hartree nonlinearity below the energy space. Using a method of randomization of functions on $\mathbb{R}^d$ associated with the Wiener decomposition, introduced by \'{A}. B\'{e}nyi, T. Oh, and O. Pocovnicu \cite{beohpo1,beohpo2}, we prove that the Cauchy problem is almost surely locally well-posed. Our result includes Hartree Schr\"odinger equation ($\alpha = 2$).