Flux-saturated porous media equations and applications

  • Juan Calvo

    Centre de Recerca Matemàtica, Bellaterra (Barcelona), Spain
  • Juan Campos

    Universidad de Granada, Spain
  • Vicent Caselles

    Universitat Pompeu-Fabra, Barcelona, Spain
  • Óscar Sánchez

    Universidad de Granada, Spain
  • Juan Soler

    Universidad de Granada, Spain


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.

Cite this article

Juan Calvo, Juan Campos, Vicent Caselles, Óscar Sánchez, Juan Soler, Flux-saturated porous media equations and applications. EMS Surv. Math. Sci. 2 (2015), no. 1, pp. 131–218

DOI 10.4171/EMSS/11