Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure

We prove global existence of weak solutions of a variant of the parabolic-parabolic Keller–Segel model for chemotaxis on the whole space \({{\mathbb {R}}^d}\) for \(d\ge 3\) with a supercritical porous-medium diffusion exponent and an external drift. The structure of the equations allow the chemotactic drift to be seen both as attraction and repulsion. The method of proof relies on the inherent gradient flow structure of this system with respect to a coupled Wasserstein- \(L^2\) metric. Additional regularity estimates are derived from the dissipation of an entropy functional.