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

Abstract : Fokker-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.
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Benoît Perthame, Alexis Vasseur. Regularization in Keller-Segel type systems and the De Giorgi method. Communications in Mathematical Sciences, International Press, 2012, 10 (2), pp.463 - 476. ⟨10.4310/CMS.2012.v10.n2.a2⟩. ⟨hal-01374730⟩

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