The Hartree–Fock equations in modulation spaces

Abstract

We establish both a local and a global well-posedness theories for the nonlinear Hartree–Fock equations and its reduced analog in the setting of the modulation spaces on ${ℝ}^d.$ In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest.

KEYWORDS AND PHRASES:

• Reduced Hartree–Fock and Hartree–Fock equations
• harmonic potential
• local and global well-posedness
• modulation spaces

2010 Mathematics Subject Classification:

• 35Q40
• 35Q55
• 42B35 (primary)
• 35A01 (secondary)

Acknowledgment

D.G. B is very grateful to Professor Kasso Okoudjou for hosting and arranging research facilities at the University of Maryland. D.G. B is thankful to SERB Indo-US Postdoctoral Fellowship (2017/142-Divyang G Bhimani) for the financial support. D.G.B is also thankful to DST-INSPIRE and TIFR CAM for the academic leave. D. G. B. is thankful to Henri Lebesgue center for current financial support. K. A. O. was partially supported by a grant from the Simons Foundation $\#319197,$ the U. S. Army Research Office grants W911NF1610008, and W911NF1910366, the National Science Foundation grant DMS 1814253, and an MLK visiting professorship.