Volume 49, Number 6, November-December 2015
Special Issue - Optimal Transport
Page(s) 1553 - 1576
Published online 05 November 2015
  1. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Clarendon Press Oxford (2000). [Google Scholar]
  2. L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lect. Math. ETH Zürich. Birkhäuser Verlag, Basel (2005). [Google Scholar]
  3. P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9 (1999) 347–359. [MathSciNet] [Google Scholar]
  4. P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Math. 66 (1994) 319–334. [Google Scholar]
  5. 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) 1–32. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  6. P. Biler, I. Guerra and G. Karch, Large global-in-time solutions of the parabolic-parabolic Keller–Segel system on the plane. Preprint arXiv:1401.7650 [math.AP] (2014). [Google Scholar]
  7. P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions. Nonlinear Anal. 23 (1994) 1189–1209. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. Blanchet and P. Laurençot, The parabolic-parabolic Keller–Segel system with critical diffusion as a gradient flow in Rd,d ≥ 3. Commun. Partial Differ. Eq. 38 (2013) 658–686. [CrossRef] [Google Scholar]
  9. A. Blanchet, V. Calvez and J.A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller–Segel model. SIAM J. Numer. Anal. 46 (2008) 691–721. [CrossRef] [MathSciNet] [Google Scholar]
  10. A. Blanchet, J.A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller–Segel model in ℝ2. Commun. Pure Appl. Math. 61 (2008) 1449–1481. [NASA ADS] [CrossRef] [EDP Sciences] [MathSciNet] [PubMed] [Google Scholar]
  11. A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. (2006) (electronic). [Google Scholar]
  12. A. Blanchet, E. Carlen and J.A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller–Segel model. J. Funct. Anal. 262 (2012) 2142–2230. [CrossRef] [MathSciNet] [Google Scholar]
  13. V. Calvez and J.A. Carrillo, Volume effects in the Keller–Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 86 (2006) 155–175. [CrossRef] [MathSciNet] [Google Scholar]
  14. V. Calvez and L. Corrias, The parabolic-parabolic Keller–Segel model in ℝ2. Commun. Math. Sci. 6 (2008) 417–447. [CrossRef] [MathSciNet] [Google Scholar]
  15. J.F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller–Segel model in the plane. Commun. Partial Differ. Equ. 39 (2014) 806–841. [CrossRef] [Google Scholar]
  16. J.A. Carrillo and F. Santambrogio, Local in time L bounds of nonlinear Fokker–Planck equations via variational schemes. In preparation (2015). [Google Scholar]
  17. J.A. Carrillo, R.J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam. 19 (2003) 1–48. [Google Scholar]
  18. J.A. Carrillo, S. Lisini and E. Mainini, Uniqueness for Keller–Segel-type chemotaxis models. Discrete Contin. Dyn. Syst. 34 (2014) 1319–1338. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  19. S. Dejak, D. Egli, P. Lushnikov and I. Sigal, On blowup dynamics in the Keller–Segel model of chemotaxis. St. Petersburg Math. J. 25 (2014) 547–574. [CrossRef] [MathSciNet] [Google Scholar]
  20. Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller–Segel chemotaxis model. SIAM J. Numer. Anal. 47 (2008) 386–408. [CrossRef] [MathSciNet] [Google Scholar]
  21. Y. Epshteyn, Upwind-difference potentials method for Patlak–Keller–Segel chemotaxis model. J. Sci. Comput. 53 (2012) 689–713. [CrossRef] [MathSciNet] [Google Scholar]
  22. M.A. Herrero and J.J. Velázquez, A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. Pisa, Cl. Sci. Serie IV 24 (1997) 633–683. [Google Scholar]
  23. S. Ibrahim, N. Masmoudi and K. Nakanishi, Trudinger–Moser inequality on the whole plane with the exact growth condition. J. Eur. Math. Soc. (JEMS) 17 (2015) 819–835. [CrossRef] [MathSciNet] [Google Scholar]
  24. R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29 (1998) 1–17. [CrossRef] [MathSciNet] [Google Scholar]
  25. E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970) 399–415. [CrossRef] [PubMed] [Google Scholar]
  26. E.F. Keller and L.A. Segel, Model for chemotaxis. J. Theor. Biol. 30 (1971) 225–234. [CrossRef] [PubMed] [Google Scholar]
  27. D. Kinderlehrer and M. Kowalczyk, The Janossy effect and hybrid variational principles. Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 153–176. [MathSciNet] [Google Scholar]
  28. D. Kinderlehrer, L. Monsaingeon and X. Xu, A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations. To appear in ESAIM: COCV (2015). Doi:10.1051/cocv/2015043. [Google Scholar]
  29. P. Laurençot and B.-V. Matioc, A gradient flow approach to a thin film approximation of the Muskat problem. Calc. Var. Partial Differ. Equ. 47 (2013) 319–341. [NASA ADS] [CrossRef] [EDP Sciences] [MathSciNet] [PubMed] [Google Scholar]
  30. D. Matthes, R.J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type. Comm. Partial Differ. Equ. 34 (2009) 1352–1397. [CrossRef] [Google Scholar]
  31. Y. Mimura, The variational formulation of the fully parabolic Keller–Segel system with degenerate diffusion. Preprint (2012). [Google Scholar]
  32. N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller–Segel system on the plane. Calc. Var. Partial Differ. Equ. 48 (2013) 491–505. [NASA ADS] [CrossRef] [EDP Sciences] [MathSciNet] [PubMed] [Google Scholar]
  33. E. Onofri, On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys. 86 (1982) 321–326. [CrossRef] [Google Scholar]
  34. F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differ. Equ. 26 (2001) 101–174. [CrossRef] [MathSciNet] [Google Scholar]
  35. P. Raphaël and R. Schweyer, On the stability of critical chemotactic aggregation. Mathematische Annalen 359 (2014) 267–377. [CrossRef] [MathSciNet] [Google Scholar]
  36. R. Schweyer, Stable blow-up dynamic for the parabolic-parabolic Patlak–Keller–Segel model. Preprint arXiv:1403.4975 [math.AP] (2014). [Google Scholar]
  37. C. Villani, Topics in optimal transportation. Vol. 58 of Grad. Stud. Math. American Mathematical Society, Providence, RI (2003). [Google Scholar]
  38. J. Zinsl, Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure. Monatshefte für Mathematik (2013) 1–27. [Google Scholar]
  39. J. Zinsl and D. Matthes, Exponential convergence to equilibrium in a coupled gradient flow system modelling chemotaxis. Anal. Partial Differ. Equ. 8 (2015) 425–466. [Google Scholar]

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