Mathematical Analysis of a Coarsening Model with Local Interactions

Abstract

We consider particles on a one-dimensional lattice whose evolution is governed by nearest-neighbor interactions where particles that have reached size zero are removed from the system. Concentrating on configurations with infinitely many particles, we prove existence of solutions under a reasonable density assumption on the initial data and show that the vanishing of particles and the localized interactions can lead to non-uniqueness. Moreover, we provide a rigorous upper coarsening estimate and discuss generic statistical properties as well as some non-generic behavior of the evolution by means of heuristic arguments and numerical observations.

Special Solutions to a Nonlinear Coarsening Model with Local Interactions

“…We collect some definitions and introduce appropriate notation to give a rigorous description of our setting. We largely follow [HNV16], where this model was first introduced in a mathematical context. Then we state our main result and give a short outline of the proof.…”

“…For negative β the existence of a sufficiently regular solution for arbitrary data cannot be expected due to G β (x) becoming singular at x = 0. Similar to the result in [HNV16], we have to restrict to initial data which satisfy a certain positivity condition:…”

“…We say that a solution u to equation (A.1) with initial data u 0 ∈ P L,d has the persistence property on [0, T ] if u(0, k) ≥ d implies u(t, k) ≥ d 2 for all t ∈ [0, T ]. By making use of the theory for the coarsening equation developed in [HNV16] we have the following result concerning Hölder regularity: Proof. First we note that because the solution satisfies u δ ≥ δ for all times then we have the Lipschitz estimate…”

“…x ≤ 2x β , (1.5) which can be integrated to yield the desired bound in the ℓ ∞ -norm. For negative β the equation givesẋ ≥ −2x β , which can be used to derive a weak upper bound, see Proposition 2.4 in [HNV16]. Furthermore, the numerical simulations and heuristics in [HNV16] demonstrate that single particles can grow linearly (thus faster than the scaling law) in On the other hand, not much is known about the validity of lower bounds.…”

“…For negative β the equation givesẋ ≥ −2x β , which can be used to derive a weak upper bound, see Proposition 2.4 in [HNV16]. Furthermore, the numerical simulations and heuristics in [HNV16] demonstrate that single particles can grow linearly (thus faster than the scaling law) in On the other hand, not much is known about the validity of lower bounds. As will be demonstrated below, there are many non-constant initial configurations which become stationary after a finite time due to the vanishing of particles.…”