In 2022, V. Giunta, T. Hillen, M. A. Lewis, J. R. Potts published a paper analyzing the well-posedness of an aggregation-diffusion equation that models multi-species interaction. The diffusion term comes from the standard Laplacian, while the aggregation term arises due to the velocity being defined as a convolution of the gradient of some kernel function against a linear combination of the density of the species. The coefficients which define this linear combination come from an interaction matrix. In this paper, the authors assume the domain is the torus with the kernel function being twice differentiable with bounded gradient. Given sufficiently smooth non-negative initial data, they establish global existence in one dimension and short-time existence in dimensions greater than or equal to two.

In this talk, we consider a wider set of kernel functions that allow singularities at the origin, the fractional Laplacian instead of the standard Laplacian, and the domain Euclidean space instead of the torus. Our kernels will be called admissible (an example of these functions is the Newtonian Potential). For admissible kernels, we establish short-time existence via a Banach-fixed point argument.

Furthermore, under extra smoothness assumptions on the kernel, we develop conditions on the interaction matrix that allow for the divergence of the velocities to remain non-negative (given non-negative initial divergence of the velocity), and then apply an inductive boot-strapping argument which establishes global existence.