Free Access
Volume 54, Number 5, September-October 2020
Page(s) 1661 - 1688
Published online 28 July 2020
  1. P. Acquistapace and B. Terreni, On quasilinear parabolic systems. Math. Ann. 282 (1988) 315–335. [Google Scholar]
  2. S. Adams, N. Dirr, M.A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Commun. Math. Phys. 307 (2011) 791. [Google Scholar]
  3. L. Alasio and S. Marchesani, Global existence for a class of viscous systems of conservation laws. Nonlinear Differ. Equ. Appl. 26 (2019) 32. [Google Scholar]
  4. L. Alasio, M. Bruna and Y. Capdeboscq, Stability estimates for systems with small cross-diffusion. ESAIM: M2AN 52 (2018) 1109–1135. [EDP Sciences] [Google Scholar]
  5. H. Amann, Dynamic theory of quasilinear parabolic systems. Math. Z. 202 (1989) 219–250. [Google Scholar]
  6. A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. Part. Differ. Equ. 26 (2001) 43–100. [Google Scholar]
  7. D. Bakry and M. Émery, Diffusions hypercontractives. In: Séminaire de Probabilités XIX 1983/84, edited by J. Azéma and M. Yor. Springer, Berlin Heidelberg, Berlin, Heidelberg (1985) 177–206. [Google Scholar]
  8. J. Berendsen, M. Burger, V. Ehrlacher and J.-F. Pietschmann, Uniqueness of strong solutions and weak–strong stability in a system of cross-diffusion equations. J. Evol. Equ. 20 (2020) 459–483. [Google Scholar]
  9. M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34 (2012) B559–B583. [Google Scholar]
  10. M. Bodnar and J.J.L. Velazquez, Derivation of macroscopic equations for individual cell-based models: a formal approach. Math. Methods Appl. Sci. 28 (2005) 1757–1779. [Google Scholar]
  11. M. Bruna and S.J. Chapman, Diffusion of multiple species with excluded-volume effects. J. Chem. Phys. 137 (2012) 204116. [Google Scholar]
  12. M. Bruna and S.J. Chapman, Excluded-volume effects in the diffusion of hard spheres. Phys. Rev. E 85 (2012) 011103. [Google Scholar]
  13. M. Bruna, M. Burger, H. Ranetbauer and M.-T. Wolfram, Asymptotic gradient flow structures of a nonlinear Fokker-Planck equation, Preprint arXiv:1708.07304 (2017). [Google Scholar]
  14. M. Bruna, M. Burger, H. Ranetbauer and M.-T. Wolfram, Cross-diffusion systems with excluded-volume effects and asymptotic gradient flow structures. J. Nonlinear Sci. 27 (2017) 687–719. [Google Scholar]
  15. M. Burger, M. Di Francesco, J.-F. Pietschmann and B. Schlake, Nonlinear cross-diffusion with size exclusion. SIAM J. Math. Anal. 42 (2010) 2842–2871. [Google Scholar]
  16. M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson–Nernst–Planck equations for ion flux through confined geometries. Nonlinearity 25 (2012) 961. [Google Scholar]
  17. M. Burger, S. Hittmeir, H. Ranetbauer and M.-T. Wolfram, Lane formation by side-stepping. SIAM J. Math. Anal. 48 (2016) 981–1005. [Google Scholar]
  18. J.A. Carrillo, H. Ranetbauer and M.-T. Wolfram, Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms. J. Comput. Phys. 327 (2016) 186–202. [Google Scholar]
  19. J.A. Carrillo, F. Filbet, M. Schmidtchen, Convergence of a finite volume scheme for a system of interacting species with cross-diffusion, Preprint arXiv:1804.04385 (2018). [Google Scholar]
  20. J.A. Carrillo, Y. Huang and M. Schmidtchen, Zoology of a nonlocal cross-diffusion model for two species. SIAM J. Appl. Math. 78 (2018) 1078–1104. [Google Scholar]
  21. L. Desvillettes, T. Lepoutre, A. Moussa and A. Trescases, On the entropic structure of reaction-cross diffusion systems. Commun. Part. Differ. Equ. 40 (2015) 1705–1747. [Google Scholar]
  22. M. Di Francesco, A. Esposito and S. Fagioli, Nonlinear degenerate cross-diffusion systems with nonlocal interaction. Nonlinear Anal. 169 (2018) 94–117. [Google Scholar]
  23. E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. [Google Scholar]
  24. N. Gavish, P. Nyquist and M. Peletier, Large deviations and gradient flows for the brownian one-dimensional hard-rod system Preprint arXiv:1909.02054 (2019). [Google Scholar]
  25. Q. Han and F. Lin, Elliptic Partial Differential Equations. American Mathematical Society 1 (2011). [Google Scholar]
  26. A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems Nonlinearity 28 (2015) 1963. [Google Scholar]
  27. A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations Springer (2016). [Google Scholar]
  28. O.A. Ladyzhenskaia, V.A. Solonnikov and N.N. Ural’tseva, Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society 23 (1988). [Google Scholar]
  29. D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation. ESAIM: M2AN 48 (2014) 697–726. [CrossRef] [EDP Sciences] [Google Scholar]
  30. L.E. Payne and H.F. Weinberger, An optimal poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5 (1960) 286–292. [Google Scholar]
  31. B. Perthame, Parabolic Equations in Biology. Springer (2015). [Google Scholar]
  32. M.J. Simpson, K.A. Landman and B.D. Hughes, Multi-species simple exclusion processes. Phys. A: Stat. Mech. Appl. 388 (2009) 399–406. [Google Scholar]
  33. G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. Springer Science & Business Media (2013). [Google Scholar]
  34. N. Zamponi and A. Jüngel, Analysis of degenerate cross-diffusion population models with volume filling. Ann. Inst. Henri Poincaré C, Anal. non linéaire 34 (2017) 1–29. [Google Scholar]
  35. W.P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer Science & Business Media 120 (1989). [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