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

By Vincent Calvez, José Antonio Carrillo. 2010

52 citations recorded by the source corpus.

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 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 case 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.

Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/673db79940d8f26656c7db912b623d02df4ae00c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

In Modeling and Simulation

Citations in the corpus

Equilibria of homogeneous functionals in the fair-competition regime (2016, background): “Hence, χc = 1 is the critical parameter in the logarithmic case that was analysed in [12] in one dimension and radial initial data in two dimensions.”
Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case (2013, background): “22) has been proved in [8, 5], see also [15, 16, 17].”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “Similar to [15], we apply the Gagliardo–Nirenberg inequality”
Asymptotic Estimates for the Parabolic-Elliptic Keller-Segel Model in the Plane (2012, result): “This result is consistent with the recent results of [11] for the two-dimensional radial model and its one-dimensional counterpart (see Proposition 16 for more comments).”