A hybrid variational principle for the Keller–Segel system in ℝ²

We construct weak global in time solutions to the classical Keller–Segel system describing cell movement by chemotaxis in two dimensions when the total mass is below the established critical value. Our construction takes advantage of the fact that the Keller–Segel system can be realized as a gradient flow in a suitable functional product space. This allows us to employ a hybrid variational principle which is a generalisation of the minimizing implicit scheme for Wasserstein distances introduced by [R. Jordan, D. Kinderlehrer and F. Otto, SIAM J. Math. Anal. 29 (1998) 1–17].