Almost sure local wellposedness of energy critical fractional Schrödinger equations with hartree nonlinearity

Abstract

We consider a Cauchy problem of energy-critical fractional Schrödinger 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 Á. Bényi, T. Oh, and O. Pocovnicu \cite{beohpo1,beohpo2}, we prove that the Cauchy problem is almost surely locally well-posed. Our result includes Hartree Schrödinger equation ($\alpha =2$).

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