Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities

Calvez, Vincent; Carrillo, José

doi:10.1090/S0002-9939-2012-11306-1

Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities
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by Vincent Calvez and José Antonio Carrillo PDF

Proc. Amer. Math. Soc. 140 (2012), 3515-3530 Request permission

Abstract:

We analyze the rate of convergence towards self-similarity for the subcritical Keller-Segel system in the radially symmetric two-dimensional case and in the corresponding one-dimensional case for logarithmic interaction. We measure convergence in the Wasserstein distance. The rate of convergence towards self-similarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the Logarithmic Hardy-Littlewood-Sobolev inequality in the one-dimensional and radially symmetric two-dimensional cases based on optimal transport arguments. In addition we prove that the one-dimensional equation is a contraction with respect to Fourier distance in the subcritical case.

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