On particle systems and critical strengths of general singular interactions

TL;DR

This paper proves global weak well-posedness for finite particle systems with singular interactions up to a critical attraction strength, and provides regularity results like heat kernel bounds near strongly attracting particles.

Contribution

It introduces a novel analytic approach using a variant of De Giorgi's method and an abstract desingularization theorem to analyze singular particle interactions.

Findings

Global weak well-posedness up to critical attraction strength

Heat kernel bounds near strongly attracting particles

Regularity results for particle systems with singular interactions

Abstract

For finite interacting particle systems with strong repulsing-attracting or general interactions, we prove global weak well-posedness almost up to the critical threshold of the strengths of attracting interactions (independent of the number of particles), and establish other regularity results, such as a heat kernel bound in the regions where strongly attracting particles are close to each other. Our main analytic instruments are a variant of De Giorgi's method in $L^p$ with appropriately chosen large $p$, and an abstract desingularization theorem.