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International audienceFokker-Planck systems modeling chemotaxis, haptotaxis and angiogenesis are numerous and have been widely studied. Several results exist that concern the gain of L p integrability but methods for proving regularizing effects in L ∞ are still very few. Here, we consider a special example, related to the Keller-Segel system, which is both illuminating and singular by lack of diffusion on the second equation (the chemical concentration). We show the gain of L ∞ integrability (strong hypercontractivity) when the initial data belongs to the scale-invariant space. Our proof is based on De Giorgi's technique for parabolic equations. We present this technique in a formalism which might be easier that the usual iteration method. It uses an additional continuous parameter and makes the relation to kinetic formulations for hyperbolic conservation laws

Regularization in Keller-Segel type systems and the De Giorgi method