Compactness criteria for constructing weak solutions without lower bounds in the fast diffusion regime

Investigate whether robust compactness criteria can be established to construct weak solutions to the nonlocal conservation law ∂_t u − div(u^m ∇g ∗ u) = 0 in cases where no positive lower bound on initial data is assumed for m < 1. Develop appropriate compactness frameworks that work despite the lack of hypoellipticity and possible degeneracies.

Background

The authors build entropy solutions via a vanishing‑viscosity scheme relying on strong compactness tools tailored to continuity equations. This approach benefits from positivity in the fast diffusion regime (m < 1). Without a lower bound, compactness becomes delicate because the equation is not hypoelliptic and shocks may form.

A general compactness criterion would broaden existence results and clarify which assumptions are truly necessary in the fast diffusion setting.