Eliseo Luongo 

Starting from the seminal work of H. Jia and V. Šverák (2015), it is now well understood that the existence of linearly unstable forward self-similar solutions can be used to establish non-uniqueness of Leray–Hopf solutions to the unforced incompressible 3D Navier--Stokes equations. Several works have since built upon the ideas of Jia and Šverák to prove non-uniqueness for various fluid dynamics models with non-zero forcing terms and, very recently, also for the unforced equations via computer-assisted methods. In this talk, we consider the parabolic–elliptic Keller–Segel equation in the mass-supercritical regime, in dimensions $3leq dleq 9$, and rigorously implement the Jia--Šverák mechanism to establish non-uniqueness of local mild solutions in the full range $1leq q<d/2$ of supercritical Lebesgue spaces. In particular, we give a fully rigorous, pen-and-paper proof of the spectral instability condition underlying the Jia--Šverák mechanism. This is joint work with U. Pappalettera.