Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory

TL;DR

This paper rigorously analyzes the conditions under which white dwarf models based on Hartree and Hartree-Fock equations undergo finite-time gravitational collapse, confirming physical theories about stellar mass limits.

Contribution

It proves finite-time blow-up for certain initial conditions, establishes local well-posedness, and confirms the Chandrasekhar limit within a rigorous mathematical framework.

Findings

Radially symmetric negative energy configurations lead to blow-up.

Solutions with small initial data exist globally in time.

Results align with Chandrasekhar's physical theory of white dwarf collapse.

Abstract

We study finite-time blow-up for pseudo-relativistic Hartree- and Hartree-Fock equations, which are model equations for the dynamical evolution of white dwarfs. In particular, we prove that radially symmetric initial configurations with negative energy lead to finite-time blow-up of solutions. Furthermore, we derive a mass concentration estimate for radial blow-up solutions. Both results are mathematically rigorous and are in accordance with Chandrasekhar's physical theory of white dwarfs, stating that stellar configurations beyond a certain limiting mass lead to ``gravitational collapse'' of these objects. Apart from studying blow-up, we also prove local well-posedness of the initial-value problem for the Hartree- and Hartree-Fock equations underlying our analysis, as well as global-in-time existence of solutions with sufficiently small initial data, corresponding to white dwarfs whose stellar mass is below the Chandrasekhar limit.