EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 49 / No 6 (November-December 2015) ESAIM: M2AN, 49 6 (2015) 1553-1576 References
Issue
ESAIM: M2AN
Volume 49, Number 6, November-December 2015
Special Issue - Optimal Transport
Page(s) 1553 - 1576
DOI https://doi.org/10.1051/m2an/2015021
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. [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. [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. [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. [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. [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. [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. [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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you