Reaction-Transport Equations with Inertia

The diffusion equation has the well-known unrealistic feature that localized disturbances spread infinitely fast, though with heavy attenuation, through the system. To see this, consider the fundamental solution (2.2). No matter how small t and how large x, the density p will be nonzero, though exponentially small. In many cases, this pathology of the diffusion equation and the reaction-diffusion reaction has negligible consequences, and (2.1) and (2.3) have proven to be satisfactory descriptions in numerous circumstances and systems. 

The origin of the unphysical behavior of the diffusion equation and the reaction-diffusion equation can be understood from three different viewpoints (i) the mathematical viewpoint, (ii) the macroscopic or phenomenological viewpoint, and (iii) the mesoscopic viewpoint. 

---

As shown above, the standard diffusion equation (2.1) has a fractional diffusion equation (2.59) as its analog in the subdiffusive case. As in the case of reaction-transport equation with inertia, see Sect. 2.2, the question arises how to combine reactions and subdiffusion in the activation-controlled regime. (For a discussion of the subdiffusion-limited case, which is outside the scope of this monograph as mentioned on page 34, see for example [491-493, 369, 391, 392, 389, 409, 410, 390, 411, 203, 187].) In some schemes, [188, 189, 186, 187], reactions terms are simply added to the fractional diffusion equation, in a manner similar to the ad hoc HRDEs (2.16), assuming at the outset that the effects of subdiffusion and reactions are separable as in the standard reaction-diffusion (2.11). However, it is easy to... 

Standard reaction-diffusion equation and introduce two deviations from normal diffusion, namely transport with inertia and anomalous diffusion. We present a phenomenological approach of standard diffusion, transport with inertia, and anomalous diffusion. This chapter also contains a first mesoscopic description of the transport in terms of random walk models. We strongly recommend such a mesoscopic approach to ensure that the reaction-transport equations studied are physically and mathematically sound. We present a comprehensive review of the mesoscopic foundations of reaction-transport equations in Chap. 3, which is at the heart of Part I. 

---