TY - UNPB

T1 - Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

AU - Collot, Charles

AU - Ghoul, Tej-Eddine

AU - Masmoudi, Nader

AU - Nguyen, Van Tien

N1 - 33 pages

PY - 2021/12/31

Y1 - 2021/12/31

N2 - We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a ring that shrinks to a point. In particular, the singularity is of type II, non self-similar. We show the stability of this dynamics among spherically symmetric solutions. In renormalised variables, the solution ressembles a traveling wave imploding at the origin, and this, to our knowledge, is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid.

AB - We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a ring that shrinks to a point. In particular, the singularity is of type II, non self-similar. We show the stability of this dynamics among spherically symmetric solutions. In renormalised variables, the solution ressembles a traveling wave imploding at the origin, and this, to our knowledge, is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid.

KW - math.AP

M3 - Preprint

BT - Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

ER -