Existence of Solutions for time-dependent fractional Kohn-Sham Equations

Abstract

We consider time-dependent Kohn-Sham equations in dimension $3$ with a fractional dispersion relation $(1-\Delta {)}^s$, $s\in (0,\frac{3}{2})$, and a class of interaction terms including, in particular, external potentials, internal potentials associated to Hartree-type non-linearities, and exchange terms described by energy subcritical pure-power non-linearities. We prove the local existence of weak solutions in $H^s$ using an approximation procedure regularizing the non-linearities. Assuming that the interaction energies can be controlled by the kinetic energy, we show that the solutions can be extended to global solutions using energy estimates. If $s\in [1,\frac{3}{2})$, we establish in addition the well-posedness of the time-dependent Kohn-Sham equations using Strichartz estimates.