Upper bounds on the coarsening rate of discrete, ill‐posed nonlinear diffusion equations

Abstract

We prove a weak upper bound on the coarsening rate of the discrete-in-space version of an ill-posed, nonlinear diffusion equation. The continuum version of the equation violates parabolicity and lacks a complete well-posedness theory. In particular, numerical simulations indicate very sensitive dependence on initial data. Nevertheless, models based on its discrete-in-space version, which we study, are widely used in a number of applications, including population dynamics (chemotactic movement of bacteria), granular flow (formation of shear bands), and computer vision (image denoising and segmentation). Our bounds have implications for all three applications.

Radial weak solutions for the Perona–Malik equation as a differential inclusion

“…Even under the lowering of regularity have enormous difficulties occurred in the existence of weak solutions. Among many different ap-proaches and attempts in this direction, e.g., the Γ-limit method in Bellettini & Fusco [3], the Young measure solutions in Chen & Zhang [4], and numerical scheme analyses in Esedoglu [9] and Esedoglu & Greer [10], to our best knowledge, Zhang [30] was the first to successfully prove that, for n = 1, there are infinitely many (Lipschitz) weak solutions to (1.1) for any given smooth initial data u 0 . His method uses the variational technique of differential inclusion together with the so called in-approximation method or convex integration; this new method can also deal with other ill-posed forward-backward diffusion problems (see, e.g., the pioneering work of Höllig [16] and its recent generalization by Zhang [31].)…”

Special Solutions to a Nonlinear Coarsening Model with Local Interactions

“…The first important a-priori estimate is the comparison principle: Proof. The proof is very similar to Lemma 2 in [EG09] and a standard maximum principle argument. Because u 0 ≥ δ, standard ODE theory gives the existence and uniqueness of a smooth solution u on the time interval [0, t * ] to equation (A.1) with δ/2 ≤ u(t, .)…”

“…The monotonicity properties of F are crucial for the qualitative behaviour of solutions and depend on the application, as an increasing flux function will lead to mass diffusion and a decreasing flux function will lead to aggregation and coarsening. A combination of both is also possible, for example in models that were investigated in [ES08,EG09].…”

Mathematical Analysis of a Coarsening Model with Local Interactions

“…and such a structure has proved successful for obtaining upper coarsening bounds in similar (finite) systems; see for instance [EG09,ES08] for the application of [KO02] to a discrete H −1 gradient flow. However, besides the fact that energy and metric tensor of infinite systems are in general infinite, this gradient flow structure does not seem useful here, since (even in finite systems) at each vanishing time the local correlations in the metric tensor change and for β ≥ 1 the energy tends to negative infinity.…”