The parabolic-parabolic Keller-Segel equation

Carrapatoso, Kleber

Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 18, 17 p.

I present in this note recent results on the uniqueness and stability for the parabolic-parabolic Keller-Segel equation on the plane, obtained in collaboration with S. Mischler in [11].

[1] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), no. 1, 213–242. | MR | Zbl

[2] M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal. 128 (1994), no. 4, 329–358. | MR | Zbl

[3] P. Biler, L. Corrias, and J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model of chemotaxis, J. Math. Biol. 63 (2011), no. 1, 1–32. | MR | Zbl

[4] P. Biler, I. Guerra, and G. Karch, Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane, . | arXiv

[5] A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations (2006), No. 44, 1–33. | MR | Zbl

[6] H. Brezis, Remarks on the preceding paper by M. Ben-Artzi: “Global solutions of two-dimensional Navier-Stokes and Euler equations” [Arch. Rational Mech. Anal. 128 (1994), no. 4, 329–358; MR1308857 (96h:35148)], Arch. Rational Mech. Anal. 128 (1994), no. 4, 359–360. | MR | Zbl

[7] V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in ${ℝ}^2$, Commun. Math. Sci. 6 (2008), no. 2, 417–447. | MR | Zbl

[8] J. Campos and J. Dolbeault, A functional framework for the Keller-Segel system: logarithmic Hardy-Littlewood-Sobolev and related spectral gap inequalities, C. R. Math. Acad. Sci. Paris 350 (2012), no. 21-22, 949–954. | MR | Zbl

[9] J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane, 2012.

[10] E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on $S^n$, Geom. Funct. Anal. 2 (1992), no. 1, 90–104. | MR | Zbl

[11] K. Carrapatoso and S. Mischler, Uniqueness and long time asymptotic for the parabolic-parabolic Keller-Segel equation, . | arXiv

[12] J. A. Carrillo, S. Lisini, and E. Mainini, Uniqueness for Keller-Segel-type chemotaxis models, Discrete Contin. Dyn. Syst. 34 (2014), no. 4, 1319–1338. | MR | Zbl

[13] L. Corrias, M. Escobedo, and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane, J. Diff. Equations 257 (2014), no. 6, 1840–1878, . | DOI | MR | Zbl

[14] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. | MR | Zbl

[15] G. Egaña and S. Mischler, Uniqueness and long time assymptotic for the Keller-Segel equation - Part I. The parabolic-elliptic case, . | arXiv

[16] L. C. F. Ferreira and J. C. Precioso, Existence and asymptotic behaviour for the parabolic-parabolic Keller-Segel system with singular data, Nonlinearity 24 (2011), no. 5, 1433–1449. | MR | Zbl

[17] N. Fournier, M. Hauray, and S. Mischler, Propagation of chaos for the 2d viscous vortex model, . | arXiv | MR

[18] M. P. Gualdani, S. Mischler, and C. Mouhot, Factorization of non-symmetric operators and exponential $H$-Theorem, . | arXiv

[19] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 4, 633–683 (1998). | Numdam | MR | Zbl

[20] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415. | Zbl

[21] S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the Kinetic-Fokker-Planck equation, . | arXiv

[22] S. Mischler and C. Mouhot, Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres, Comm. Math. Phys. 288 (2009), no. 2, 431–502. | MR | Zbl

[23] S. Mischler and J. Scher, Semigroup spectral analysis and growth-fragmentation equation, to appear in Ann. Inst. H. Poincaré - Anal. Non Linéaire (2015).

[24] T. Nagai, Global existence and blowup of solutions to a chemotaxis system, Proceedings of the Third World Congress of Nonlinear Analysts, Part 2 (Catania, 2000), vol. 47, 2001, pp. 777–787. | MR | Zbl

[25] T. Nagai, T. Senba, and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J. 30 (2000), no. 3, 463–497. | MR | Zbl

[26] Y. Naito, T. Suzuki, and K. Yoshida, Self-similar solutions to a parabolic system modeling chemotaxis, J. Differential Equations 184 (2002), no. 2, 386–421. | MR | Zbl

[27] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311–338. | MR | Zbl

[28] I. Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, . | arXiv