Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

Abstract

Phase-field models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen-Cahn equation with a potential with N ≥ 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow.In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: As long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen-Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter ε 0. We even establish the rate of convergence O(ε 1/2 ). Our approach is based on the gradient flow structure of the Allen-Cahn equation and its limiting motion: Building on the recent concept of "gradient flow calibrations" for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen-Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e. g. avoiding the need for a stability analysis of the Allen-Cahn operator or additional convergence hypotheses for the energy at positive times.

Weak-strong uniqueness for volume-preserving mean curvature flow

“…This guarantees that on Σ * (t), ( 9) simply reduces to (12). By the Lipschitz continuity of all functions appearing on the left-hand side of ( 9), this implies the validity of (9). Item (vi) follows directly from the construction of ϑ.…”

“…Item (iii) follows directly from the construction of ξ. The evolution equation (9) in Item (v) is also built into the construction of B, namely through the boundary condition (26). This guarantees that on Σ * (t), ( 9) simply reduces to (12).…”

“…We start with the two terms which have to be handled differently than in the case of vanilla mean curvature flow: the fourth term (λ * − λ)B • ν|∇χ| looks rather worrying and seems to require a stability analysis for the Lagrange-multipliers λ and λ * . However, since by ( 4), (9), and (10)…”

“…Liu and the author [21] combined the relative entropy method with weak convergence methods to derive the sharp-interface dynamics of isotropic-nematic phase transitions in liquid crystals. Most recently, Fischer and Marveggio [9] extended the result [8] to the vector-valued Allen-Cahn equation and proved its convergence to multiphase mean curvature flow. Previous to this result, only formal arguments [4] and the conditional result [23] were known.…”

Diffuse-interface approximation and weak–strong uniqueness of anisotropic mean curvature flow

“…A related approach regarding a localized energy excess was introduced by the first author with Otto in [41] where they proved convergence of the MBO thresholding scheme to multi-phase mean curvature flow. A nontrivial modification of this idea -based on controlling the tilt excess of the sharp or diffuse interface with regard to a smooth approximation -has been used to prove convergence of the vectorial Allen-Cahn equation to multiphase mean curvature flow [42] and derive an associated rate of convergence [25] along with optimal quantitative convergence rates for the Allen-Cahn equation to mean curvature flow in the two-phase setting [24]. A key feature of viscosity type solutions is the associated comparison principle, which automatically provides uniqueness of solutions up to the issue of fattening.…”