Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation

By Eric A. Carlen, Alessio Figalli. 2011

125 citations recorded by the source corpus.

Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several dierent stability results for a Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the connection between this inequality and a fast diusion 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/2cab67afaf0c771214cc04deb3ba750fe049d5f7

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

The quantitative isoperimetric inequality and related topics (2015, methodology): “Finally, a quantitative version of the Gagliardo–Sobolev–Nirenberg inequality, has been recently proved by Carlen and Figalli in [35], starting from the Bianchi and Egnell stability result that we have discussed in Section 6.”
Equilibria of homogeneous functionals in the fair-competition regime (2016, background): “The critical case χ = 1 was analysed further in [8,6,16] in terms of stability of stationary states.”
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher (2011, methodology): “(14) toward a free energy solution to the Patlak-Keller-Segel system (1). The proof of the asymptotics result is then a standard entropy method as set out at the end of Section 2.3. Very recently, in [20], E. Carlen and A. Figalli use a argument of Bianchi-Egnell to obtain a quantitative stability f”
Remainder terms in the fractional Sobolev inequality (2012, background): “, [10, 11, 17]); see also [7] for a recent application of remainder terms.”
Sharp gradient stability for the Sobolev inequality (2020, background): “Introduction Motivated by important applications to problems in the calculus of variations and evolution PDEs, in recent years there has been a growing interest around the understanding of quantitative stability for functional/geometric inequalities, see for instance [3, 2, 8, 27, 28, 21, 9, 22, 29,”
Duality and Stability for Functional Inequalities (2016, background): “Quantitative stability for the Logarithmic HLS inequality under additional assumptions on f has been proved in [6], the additional assumptions are natural for the application made there to the Keller–Segel equation, but the application would be strengthened by relaxing them.”