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.

References

However, even if we assume positive initial data, the existence of weak solutions remains open in the case $m<1$.

— On a repulsion model with Coulomb interaction and nonlinear mobility  (2510.16894 - Courcel et al., 19 Oct 2025) in Subsection “Possible extensions” (Introduction)

We do not address the more challenging case $s<1$, as it is currently unclear how to derive suitable discrete chain-rule inequalities in this regime.

— Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules  (2609.09921 - Hirvonen et al., 9 Sep 2026) in Section 1, subsection “Main results”