Asymptotic behavior near planar transition fronts for the Cahn–Hilliard equation

Abstract

We consider the asymptotic behavior of perturbations of planar wave solutions arising in the CahnHilliard equation in space dimensions d ≥ 2. Such equations are well known to arise in the study of spinodal decomposition, a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into regions in which one component or the other is dominant, with these regions separated by steep transition layers. A critical feature of the Cahn-Hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis, a feature that is known to complicate the step from spectral to nonlinear stability. The analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator (leading in the case of stability) moves into the negative-real half plane with cubic scaling λ ∼ |ξ| 3 , where ξ ∈ R d−1 denotes a Fourier variable associated with spatial components transverse to the planar wave. Under the assumption of spectral stability, described in terms of an appropriate Evans function, we develop detailed asymptotics for perturbations from planar wave solutions, establishing asymptotic stability for initial perturbations decaying with appropriate algebraic rate in an L 1 norm of the transverse variables.

Spectral analysis of planar transition fronts for the Cahn–Hilliard equation

“…Theorem 1.2 verifies conditions (1) and (2) from p. 128 of [8]. In [8] it is shown that if these conditions hold then a nonlinear iteration can be closed on an appropriate integral equation for a perturbation function v(t, x) defined by…”

“…As observed in [8] (see also [17,18]), this Evans function is not analytic in a neighborhood of (λ, ξ ) = (0, 0). More precisely, the functions φ − 1 (x 1 ; λ, ξ ) and φ + 2 (x 1 ; λ, ξ ) fail to be analytic in this neighborhood.…”

“…Letting φ − 1 (x 1 ; λ, ξ ) and φ − 2 (x 1 ; λ, ξ ) denote the two linearly independent asymptotically decaying solutions at −∞ of (1.8) (for λ away from essential spectrum), and φ + 1 (x 1 ; λ, ξ ) and φ + 2 (x 1 ; λ, ξ ) similarly the two linearly independent asymptotically decaying solutions at +∞ (this decomposition is established in Lemma 3.1 of [8]), we note that the eigenfunction φ(x 1 ; λ, ξ ) must be a linear combination of φ − 1 (x 1 ; λ, ξ ) and φ − 2 (x 1 ; λ, ξ ) and also of φ + 1 (x 1 ; λ, ξ ) and φ + 2 (x 1 ; λ, ξ ). In this way, we only have an eigenvalue if there is linear dependence among these four solutions; that is, if…”

“…This theorem is an immediate consequence of Theorem 1.1 of [7]. (The nonlinear terms dropped off in this linearization will not be considered here, but details of the nonlinear analysis can be found in [8].) Observing that the coefficients of L depend only on the distinguished variable x 1 , we take a Fourier transform in the transverse coordinatesx := (x 2 , x 3 , .…”

“…is the asymptotic decay rate, given by [8] or Section 3 of the current paper.) Clearly, μ − 3 (λ, ξ ) is not analytic in any neighborhood of (0, 0), and the Evans function inherits this lack of analyticity.…”