Free Access
Issue
ESAIM: COCV
Volume 21, Number 2, April-June 2015
Page(s) 414 - 441
DOI https://doi.org/10.1051/cocv/2014032
Published online 04 March 2015
  1. L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Lect. Math. ETH Zürich, 2nd edition. Birkhäuser Verlag, Basel (2008). [Google Scholar]
  2. D. Balagué, J.A. Carrillo, T. Laurent and G. Raoul, Dimensionality of local minimizers of the interaction energy. Arch. Ration. Mech. Anal. 209 (2013) 1055–1088. [CrossRef] [Google Scholar]
  3. D. Balague, J.A. Carrillo, T. Laurent and G. Raoul, Nonlocal interactions by repulsive-attractive potentials: radial ins/stability. Phys. D 260 (2013) 5–25. [CrossRef] [MathSciNet] [Google Scholar]
  4. A.L. Bertozzi and J. Brandman, Finite-time blow-up of L-weak solutions of an aggregation equation. Commun. Math. Sci. 8 (2010) 45–65. [CrossRef] [Google Scholar]
  5. A.L. Bertozzi, J.A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22 (2009) 683–710. [CrossRef] [Google Scholar]
  6. A.L. Bertozzi and T. Laurent, The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels. Chin. Ann. Math. Ser. B 30 (2009) 463–482. [CrossRef] [MathSciNet] [Google Scholar]
  7. A.L. Bertozzi, T. Laurent and J. Rosado, Lp theory for the multidimensional aggregation equation. Comm. Pure Appl. Math. 64 (2011) 45–83. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional keller-segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Eq. 44 (2006). [Google Scholar]
  9. F. Bolley, Y. Brenier and G. Loeper, Contractive metrics for scalar conservation laws. J. Hyperbolic Differ. Eq. 2 (2005) 91–107. [CrossRef] [Google Scholar]
  10. G.A. Bonaschi, Gradient flows driven by a non-smooth repulsive interaction potential. Master’s thesis, University of Pavia, Italy (2011). Preprint arXiv:1310.3677. [Google Scholar]
  11. Y. Brenier, L2 formulation of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal. 193 (2009) 1–19. [CrossRef] [Google Scholar]
  12. Y. Brenier, W. Gangbo, G. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions. J. Math. Pures Appl. 99 (2013) 577–617. [CrossRef] [Google Scholar]
  13. A. Bressan, Global solutions of systems of conservation laws by wave-front tracking. J. Math. Anal. Appl. 170 (1992) 414–432. [CrossRef] [Google Scholar]
  14. A. Bressan, Hyperbolic systems of conservation laws, The one-dimensional Cauchy problem. In vol. 20 of Oxford Lect. Ser. Math. Appl. Oxford University Press, Oxford (2000). [Google Scholar]
  15. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. In vol. 5 of Math. Stud. Notas Mat. Publishing Co., Amsterdam (1973). [Google Scholar]
  16. M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion. Netw. Heterog. Media 3 (2008) 749–785. [CrossRef] [MathSciNet] [Google Scholar]
  17. J.A. Carrillo, Y.P. Choi and M. Hauray, The derivation of Swarming models: Mean-Field Limit and Wasserstein distances. Collective Dynamics From Bacteria to Crowds, vol. 553 of CISM (2014) 1–46. [Google Scholar]
  18. J.A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156 (2011) 229–271. [CrossRef] [MathSciNet] [Google Scholar]
  19. J.A. Carrillo, L.C.F. Ferreira and J.C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity. Adv. Math. 231 (2012) 306–327. [CrossRef] [MathSciNet] [Google Scholar]
  20. J.A. Carrillo, S. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials. Nonlinear Anal. Theor. Methods Appl. 100 (2014) 122–147. [CrossRef] [Google Scholar]
  21. C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972) 33–41. [CrossRef] [MathSciNet] [Google Scholar]
  22. M. Di Francesco and D. Matthes, Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations. Calc. Var. Part. Differ. Eq. 50 (2014) 199–230. [CrossRef] [Google Scholar]
  23. R.J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J. Differ. Eq. 20 (1976) 187–212. [CrossRef] [Google Scholar]
  24. R.L. Dobrušin, Vlasov equations. Funktsional. Anal. i Prilozhen. 13 (1979) 48–58, 96. [CrossRef] [MathSciNet] [Google Scholar]
  25. L.C. Evans, Partial differential equations. In vol. 19 of Grad. Stud. Math. American Mathematical Society, Providence, RI (1998). [Google Scholar]
  26. K.Fellner and G. Raoul, Stable stationary states of non-local interaction equations. Math. Models Methods Appl. Sci. 20 (2010) 2267–2291. [CrossRef] [Google Scholar]
  27. K. Fellner and G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials. Math. Comput. Model. 53 (2011) 1436–1450. [CrossRef] [Google Scholar]
  28. R.C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24 (2011) 2681–2716. [CrossRef] [Google Scholar]
  29. N. Gigli and F. Otto, Entropic burgers’ equation via a minimizing movement scheme based on the wasserstein metric. Calc. Var. Partial Differ. Eq. 1–26 (2012). [Google Scholar]
  30. F. Golse, The mean-field limit for the dynamics of large particle systems. In Journées “Équations aux Dérivées Partielles”, pages Exp. No. IX, 47. Univ. Nantes, Nantes (2003). [Google Scholar]
  31. W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329 (1992) 819–824. [Google Scholar]
  32. 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]
  33. S.N. Kružkov, First order quasilinear equations in serveral independent variables. Math. USSR Sb 10 (1970) 217–243. [Google Scholar]
  34. S.N. Kružkov, Generalized solutions of the Cauchy problem in the large for first order nonlinear equations. Dokl. Akad. Nauk. SSSR 187 (1969) 29–32. [Google Scholar]
  35. P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The theory of classical and nonclassical shock waves. Springer (2002). [Google Scholar]
  36. H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows. Arch. Ration. Mech. Anal. 172 (2004) 407–428. [CrossRef] [Google Scholar]
  37. R.J. McCann, A convexity principle for interacting gases. Adv. Math. 128 (1997) 153–179. [CrossRef] [MathSciNet] [Google Scholar]
  38. A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm. J. Math. Biol. 38 (1999) 534–570. [CrossRef] [Google Scholar]
  39. L. Natile and G. Savaré, A Wasserstein approach to the one-dimensional sticky particle system. SIAM J. Math. Anal. 41 (2009) 1340–1365. [CrossRef] [MathSciNet] [Google Scholar]
  40. O.A. Oleinik, Discontinuous solutions of nonlinear differential equations. Amer. Math. Soc. Transl. 26 (1963) 95–172. [MathSciNet] [Google Scholar]
  41. C.S. Patlak, Random walk with persistence and external bias. Bull. Math. Biophys. 15 (1953) 311–338. [CrossRef] [Google Scholar]
  42. D. Serre, Systems of conservation laws. 1. Hyperbolicity, entropies, shock waves. Translated from the 1996 French original by I.N. Sneddon. Cambridge University Press, Cambridge (1999). [Google Scholar]
  43. C.M. Topaz, A.L. Bertozzi and M.E. Lewis, A nonlocal continuum model for biological aggregations. Bull. Math. Biol. 68 (2006) 1601–1623. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  44. C.J. Van Duijn, L.A. Peletier and I.S. Pop, A new class of entropy solutions of the buckley-leverett equation. SIAM J. Math. Anal. 39 (2007) 507–536. [CrossRef] [MathSciNet] [Google Scholar]
  45. C. Villani, Topics in optimal transportation. In vol. 58 of Grad. Stud. Math. American Mathematical Society, Providence, RI (2003). [Google Scholar]
  46. G.B. Whitham, Linear and nonlinear waves. John Wiley & Sons (1974). [Google Scholar]

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