Existence of weak solutions for the two‑species interaction system when m < 1

Establish existence of weak solutions for the two‑population system ∂_t u_1 − div(u_1^m ∇g ∗ (u_1 + u_2)) = 0 and ∂_t u_2 − div(u_2^m ∇g ∗ (u_1 + u_2)) = 0 on the d‑dimensional torus in the fast diffusion regime m < 1, even for positive initial data. Clarify whether positivity and gradient‑flow structure suffice to obtain global weak solutions.

Background

The proposed two‑species generalization couples each population to a common nonlocal pressure generated by the total density but with distinct nonlinear mobilities. For m < 1, the single‑species model admits a gradient‑flow formulation, suggesting potential existence results. However, maintaining positivity and constructing solutions for the coupled system present significant challenges.

Resolving existence for m < 1 would extend the analytical framework to multi‑species interactions and support modeling applications.