On Blowup for time-dependent generalized Hartree-Fock equations

TL;DR

This paper proves finite-time blowup for certain attractive fermionic systems modeled by Hartree-Fock and Hartree-Fock-Bogoliubov equations, extending previous results to more general cases and including angular momentum considerations.

Contribution

It extends previous blowup results to Hartree-Fock equations with infinite rank solutions and general Newtonian interactions, and demonstrates blowup in a Hartree-Fock-Bogoliubov model with angular momentum cutoff.

Findings

Finite-time blowup for spherically symmetric negative energy solutions.

Extension of blowup results to infinite rank Hartree-Fock solutions.

Existence of blowup in a Hartree-Fock-Bogoliubov model with angular momentum cutoff.

Abstract

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree-Fock-Bogoliubov theory.