Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation
نویسندگان
چکیده
Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the connection between this inequality and a fast diffusion equation, we get stability for the Log-HLS inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller-Segel system.
منابع مشابه
Keller-Segel, Fast-Diffusion and Functional Inequalities
We will show how the critical mass classical Keller-Segel system and the critical displacement convex fast-diffusion equation in two dimensions are related. On one hand, the critical fast diffusion entropy functional helps to show global existence around equilibrium states of the critical mass Keller-Segel system. On the other hand, the critical fast diffusion flow allows to show functional ine...
متن کاملFunctional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model
We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions ̺λ, λ > 0, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional Hλ coming from the critical fast diffusion equation in R ...
متن کاملTwo-dimensional Keller-segel Model: Optimal Critical Mass and Qualitative Properties of the Solutions
The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutio...
متن کاملA finite volume scheme for a Keller-Segel model with additional cross-diffusion
A finite volume scheme for the (Patlak-) Keller-Segel model in two space dimensions with an additional cross-diffusion term in the elliptic equation for the chemical signal is analyzed. The main feature of the model is that there exists a new entropy functional yielding gradient estimates for the cell density and chemical concentration. The main features of the numerical scheme are positivity p...
متن کامل[hal-00503203, v1] Refined Asymptotics for the subcritical Keller-Segel system and Related Functional Inequalities
We analyze the rate of convergence towards self-similarity for the subcritical KellerSegel 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...
متن کامل