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 16 π , 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 ( R 2 ) . 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.