Suppressing Chemotactic Blow-Up Through a Fast Splitting Scenario on the Plane
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We revisit the question of global regularity for the Patlak–Keller–Segel (PKS) chemotaxis model. The classical 2D parabolic-elliptic model blows up for initial mass M> 8 π. We consider a more realistic scenario which takes into account the flow of the ambient environment induced by harmonic potentials, and thus retain the identical elliptic structure as in the original PKS. Surprisingly, we find that already the simplest case of linear stationary vector field, Ax ⊤ , with large enough amplitude A, prevents chemotactic blow-up. Specifically, the presence of such an ambient fluid transport creates what we call a ‘fast splitting scenario’, which competes with the focusing effect of aggregation so that ‘enough mass’ is pushed away from concentration along the x 1 -axis, thus avoiding a finite time blow-up, at least for M< 16 π. Thus, the enhanced ambient flow doubles the amount of allowable mass which evolve to global smooth solutions.
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