Pinning and de-pinning phenomena in front propagation in heterogeneous media

Abstract

This paper investigates the pinning and de-pinning phenomena of some evolutionary partial differential equations which arise in the modelling of the propagation of phase boundaries in materials under the combined effects of an external driving force F and an underlying heterogeneous environment. The phenomenology is the existence of pinning states -stationary solutions -for small values of F, and the appearance of genuine motion when F is above some threshold value. In the case of a periodic medium, we characterise quantitatively, near the transition regime, the scaling behaviour of the interface velocity as a function of F . The results are proved for a class of some semi-linear and reaction-diffusion equations.

Pinned fronts in heterogeneous media of jump type

“…For instance in the field of superconductivity, there is quite an extensive literature, dating back to the seventies, on the impact of spatially localized heterogeneities, or 'impurities', on the dynamics of localized structures (the so-called fluxons), see [14]. There is also a special interest in the influence of spatial heterogeneities on the dynamics of localized structures such as fronts and pulses in reaction-diffusion equations, see [1,2,16,18,24,25] and the references therein. The pinning phenomenon, in which a traveling solution gets trapped by the heterogeneity while its homogeneous equivalent would have kept on traveling, can be considered as one of its most dramatic effects.…”

“…For instance, scalar systems have a gradient structure, and their solutions can be controlled by sub-and super-solutions. These properties are crucial ingredients in the analysis of [2]. Moreover, unlike the models used for pinning in superconductivity, (1.1) is not close to a completely integrable partial differential equation (PDE), so that the impact of (localized, spatial) heterogeneities cannot be studied by a perturbation method based on this fact [14].…”

“…While pinning has been studied analytically in the special case of scalar equations (see [2,24] and the references therein), it has been studied much less extensively for systems of reaction-diffusion equations [9,10,11], see Remark 1.1. Numerically, pinning for systems of reaction-diffusion equations has been studied more thoroughly.…”

“…System (1.1) is also a natural extension of the FHN equations to systems with two inhibitors. Since (1.1) is a system, the pinning phenomenon cannot be studied by the methods employed in [2], which rely heavily on the scalar nature of the equations under consideration. For instance, scalar systems have a gradient structure, and their solutions can be controlled by sub-and super-solutions.…”

Numerical and analytical aspects of the pinning of martensitic phase boundaries

“…In [8] it is proved that the depinning transition of a parabolic system (Laplacian instead of the square root of the Laplacian in (1)) exhibits-under certain non-degeneracy conditions-a power law behavior. In this case, close to the critical driving force, the average velocityv, of the unique space-time periodic solution is given byv = C· √ F − F * +o( √ F − F * ), where C depends on the local forcing ϕ.…”