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

Fokker-Planck systems modeling chemotaxis, haptotaxis and angiogenesis are numerous and have been widely studied. Several results exist that concern the gain of L 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. Key-words De Giorgi method, entropy methods, Regularizing effects, Hypercontractivity, KellerSegel system, haptotaxis. AMS Class. No. 35K55, 35B65, 92C17