Contact angle selection for interfaces in growing domains

Abstract

We study interfaces in an Allen‐Cahn equation, separating two metastable states. Our focus is on a directional quenching scenario, where a parameter renders the system bistable in a half plane and monostable in its complement, with the region of bistability expanding at a fixed speed. We show that the growth mechanism selects a contact angle between the boundary of the region of bistability and the interface separating the two metastable states. Technically, we focus on a perturbative setting in a vicinity of a symmetric situation with perpendicular contact. The main difficulty stems from the lack of Fredholm properties for the linearization in translation invariant norms. We overcome those difficulties establishing Fredholm properties in weighted spaces and farfield‐core decompositions to compensate for negative Fredholm indices.

Wavenumber selection via spatial parameter jump

“…In particular, one can now ask for solutions asymptotic to stripes that are perpendicular to the parameter jump, with kx = 0, or at an oblique angle to the parameter jump. In the much simpler Allen-Cahn equation, and to some extent in the slightly more complicated Cahn-Hilliard equation, such solutions have been constructed in [30][31][32], showing in particular that stripes are either parallel or perpendicular to the parameter jump in this case. In the case of the Swift-Hohenberg equation, pursuing an infinitedimensional center-manifold and normal form analysis following the analysis of grain boundaries in [19,20,39] seems to be a promising generalization of the approach developed here.…”

Horizontal patterns from finite speed directional quenching

“…For instance, when α = 0 and c y = 0 the function Ξ (cx) ∞ (·, ·) solves (1.9) satisfying the limit (1.10) for ϕ * = π 2 and u ± = ±1. It was shown in [MS17a] shows that for any c x 0 fixed, the function Ξ (c=cx) ∞ (·, ·) can be continued in ϕ * as a solution to (1.9) for all ϕ * − π 2 sufficiently small. One of the most important properties used in their proof concerns to the strict monotonicity of the mapping y → Ξ (cx) ∞ (x, y) for any x ∈ R, namely,…”

“…Nevertheless, the authors managed to prove (1.11) for c x 0 sufficiently small (see [MS17a,Prop. 4.1]).Our construction readily gives the validity of (1.11) for all 0 < c x < 2, thus we can make use of the analysis in [MS17a] to conclude the following result:…”

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

“…Both techniques had been seen separately in earlier works but, to the authors knowledge, have never been used simultaneously. The far/near (spatial) decomposition as done in (9) has been called by some authors far field-core decomposition: it has been a building block in the construction of multidimensional patterns in extended domains [dPKPW10], in the study of perturbation effects on the far field of multidimensional patterns [MS18,Mon18]; in combination with bifurcation techniques it is present in the context of pattern formation as one can see in the work of Scheel and collaborators (see for instance [MS17,GS16], specially [LS17] and [MS15,§5]). Similar types of decompositions have also been exploited in combination with homogenization techniques in the study and simulation of micro-structures and defects [BLBL12,BLBL15].…”

“…In spite of its robustness, dynamical systems techniques do not seem to be broad enough to capture unsurmountable difficulties in the study of multidimensional patterns. Recently, different research avenues have been exploited: several studies have been done using rigorous numerical analysis [MS13], harmonic analysis techniques [JS15, Jar15, BLBL12], variational techniques [Rab94], or more functional-analytic based techniques [MS18]. Still, many classes of problems remain unsolved, as that of asymmetrical grain boundaries, a case that does not seem to be directly amenable to the spatial dynamics techniques as presented in [HS12,SW14]; in this scenario the far/near decompositions we presented might be relevant for analytical results.…”