Existence of axis-symmetric blow-up solution with multiple peak aggregations for the 2-D Keller-Segel systems coupled bipolar source and sink flow

We consider the Keller-Segel systems in $R^d$, coupled with a bipolar source and sink flow. Focusing on the two-dimensional case ($d=2$), we establish finite-time blow-up of solutions under an axis-symmetric setting, without requiring the solutions to be radial. In particular, we prove that multiple blow-up points appear in pairs (i.e., in even numbers) away from the origin, lying on the $x_1$-axis and exhibiting axis-symmetry about the $x_2$-axis. This result holds for initial data with total mass strictly greater than 16π, and stands in contrast to the classical radial setting, where blow-up is confined to the origin.

A crucial part of our analysis is a sharp ε-regularity theorem, originally developed for the classical Keller-Segel systems and first established by Luckhaus–Sugiyama–Velázquez [12]. This theorem states that if the local mass around $x_1$ is sufficiently small at some time $t_1$, then the solution remains locally bounded in a suitable parabolic cylinder in space–time centered at $(x_1,t_1)$. Compared to the classical ε-regularity theorem, it requires weaker assumptions and yields weaker conclusions, making it a form of partial regularity that is particularly essential for analyzing blow-up singularities.

Based on this sharp ε-regularity theorem, we further prove that only finitely many blow-up points appear as singular sets, and the asymptotic profile is characterized as the sum of a finite number of δ-functions and a regular part in $L^1\left(R^2\right)$. Moreover, our results reveal that multi-peak blow-up phenomena can occur with or without the presence of non-trivial flow, highlighting the intricate interplay between diffusion, chemotaxis, and persistent advection. By accounting for non-decaying flow and employing precise blow-up criteria, we establish that the blow-up time can be bounded above by any prescribed threshold. These findings are justified through the construction of a time-local existence and extension theory for strong solutions, which incorporates both advection and mass conservation.