Geodesic
Flux-saturated porous media equations and applications
Juan Calvo1 ; Juan Campos2 ; Vicent Caselles3 ; Óscar Sánchez2 ; Juan Soler2
1Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain
2Universidad de Granada, Spain
3Universitat Pompeu-Fabra, Barcelona, Spain
EMS surveys in mathematical sciences, Tome 2 (2015) no. 1, pp. 131-218

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DOI

The aim of this paper is to review the main recent results about the dynamics of nonlinear partial differential equations describing flux-saturated transport mechanisms, eventually in combination with porous media flow and/or reactions terms. The result is a system characterized by the presence of wave fronts which move defining an interface. This can be used to model different process in applications in a variety of areas as Developmental Biology or Astrophysics. The concept of solution and its properties (well-posedness in a Bounded Variation scenario, Rankine–Hugoniot and geometric conditions for jumps, regularity results, finite speed of propagation, ...), qualitative study of these fronts (traveling waves in particular) and application in morphogenesis cover the panorama of this review.
DOI : 10.4171/emss/11
Classification : 35-XX, 34-XX, 37-XX, 70-XX
Mots-clés : Flux limitation, flux-saturation, porous media equations, waiting time, pattern formation, traveling waves, optimal mass transportation, entropy solutions, relativistic heat equation, complex systems

Juan Calvo  1   ; Juan Campos  2   ; Vicent Caselles  3   ; Óscar Sánchez  2   ; Juan Soler  2

1 Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain
2 Universidad de Granada, Spain
3 Universitat Pompeu-Fabra, Barcelona, Spain 
Juan Calvo; Juan Campos; Vicent Caselles; Óscar Sánchez; Juan Soler. Flux-saturated porous media equations and applications. EMS surveys in mathematical sciences, Tome 2 (2015) no. 1, pp. 131-218. doi: 10.4171/emss/11
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     title = {Flux-saturated porous media equations and applications},
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     pages = {131--218},
     year = {2015},
     volume = {2},
     number = {1},
     doi = {10.4171/emss/11},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/emss/11/}
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