Flux-saturated porous media equations and applications

Abstract

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.

Cross-diffusion and traveling waves in porous-media flux-saturated Keller–Segel models

“…Then, exploring or modeling new nonlinear transport/diffusion phenomena is an interesting subject not only from the viewpoint of applications, but also from a mathematical perspective. In the framework of flux-saturated mechanisms, traveling waves have been studied in [15,14,17,18]. In this context, as we will show, the Keller-Segel system with flux-saturated mechanism (5), which is the object of study in this paper, exhibits new properties with respect to the classical one.…”

“…Because of the symmetry of the function G, we deduce that it is necessarily satisfied υ − = υ + := υ ± . Then, we obtain the following result: σt) is an entropic solution on the frame [14,15], then its derivative must be a Radon measure on the support. If we assume that this support is finite, then it can be shown that it is one of those obtained previously.…”

“…However, the proposed Keller-Segel model with flux-saturated mechanism will require of a more precise analysis. Other flux-saturated porous-media systems have been analyzed in [14,15]. Flux-saturated equations were introduced in [26,40,44,46,49], while the general theory for the existence of entropy solutions associated with flux-saturated equations has been widely developed in the framework of Bounded Variation functions, see [2,3,4,13,32].…”

“…Regularity properties were analyzed in [2,14,16,19]. Applications of these ideas to diverse contexts such as Physics, Astronomy or Biology can be found for instance in [14,44,49,50]. The deduction of the Keller-Segel equations from the foundations of the kinetic theory as a combination of parabolic and hyperbolic limits is also of interest, see [5,6,22,23,42,47] for the derivation of the classical Keller-Segel system and [7,8,24] in the case of flux-saturated or porous media mechanisms of spreading.…”

“…These arguments will be developed in Section 2 and have been inspired in [14,16,19]. (2) Possible Blow-up and dynamic of the support: We analyze the dynamic properties of the evolution of the support of the solution in terms of the three parameters of the system: the mass M , the flux-saturated characteristic speed c, and the chemoattractant sensitivity constant a.…”