Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies

We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen-Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.


Introduction
The Allen-Cahn equation -with a suitable double-well potential W like for instance W (s) = c (1 − s 2 ) 2 , c > 0 -is the most natural diffuse-interface approximation for (two-phase) mean curvature flow: It is well-known that in the limit of vanishing interface width ε → 0, the solutions u ε to the Allen-Cahn equation (1) converge to a characteristic function χ : R d × [0, T ] → {−1, 1} whose interface evolves by motion by mean curvature. For a proof of this fact in the framework of Brakke solutions to mean curvature flow, we refer to [8], while for the convergence towards the viscosity solution of the level-set formulation under the assumption of non-fattening we refer to [5]. Provided that the total energy converges in the limit ε → 0, one may prove that the limit is a distributional solution [10]. For a general compactness statement using the gradient-flow structure of (1) and the identification of the limit in the radially symmetric case, we refer the reader to [2]. Under the assumption of the existence of a smooth limiting evolution, rates of convergence may be derived based on a strategy of matched asymptotic expansions and the stability of the linearized Allen-Cahn operator [3,4].
The Allen-Cahn equation corresponds to the L 2 gradient flow of the Ginzburg-Landau energy functional Solutions to the Allen-Cahn equation (1) satisfy the energy dissipation estimate d dtˆRd In the present work, we pursue a strategy of deriving a quantitative convergence result in the sharp-interface limit ε → 0 based purely on the energy dissipation structure. In particular, we give a short proof for the following quantitative convergence of solutions of the Allen-Cahn equation towards a smooth solution of mean curvature flow.
, be a compact interface I(t) = ∂Ω(t) evolving smoothly by mean curvature, and let χ : Let W be a standard double-well potential as described below and denote by θ the corresponding one-dimensional interface profile. Let u ε be the solution to the Allen-Cahn equation (1) with initial data given by

holds.
We note that this error estimate is of optimal order, as ε is the typical width of the diffuse interface in the Allen-Cahn approximation (i. e. the typical width of the region in which the function ψ ε takes values in the range [−1 + δ, 1 − δ] for any fixed δ > 0).
As our quantitative convergence analysis does not rely on the comparison principle, it may be applicable to more complex models, such as systems of Navier-Stokes-Allen-Cahn type [1]; note that a weak-strong uniqueness theorem for the two-fluid free boundary problem for the Navier-Stokes equation (i. e. the corresponding sharp-interface model) has already been obtained in [6]. We note that a relative entropy concept related to the one in [6] had already been employed by Jerrard and Smets [9] to deduce weak-strong uniqueness of solutions to binormal curvature flow. In the forthcoming work [7], we employ an energy-based strategy to deduce a weak-strong uniqueness theorem for multiphase mean curvature flow.

Definition of the Relative Energy and Gronwall Estimate
2.1. Extending the unit normal vector field of the surface evolving by mean curvature. Let I = I(t) be a surface that evolves smoothly by motion by mean curvature. Fix r c > 0 small enough depending on (I(t)) t∈[0,T ] . For each t ∈ [0, T ], we extend the inner unit normal n I of the surface I(t) to a vector field on R d by defining where the map P I : R d → I = I(t) is the nearest point projection and where η is a cutoff satisfying whereη is a standard cut-off which is identically 1 in a neighborhood of 0.
The extended unit normal vector field ξ and mean curvature vector H I (x) := H I (P I x)η(dist(x, I)) then satisfy the PDEs and Furthermore, we have the estimate To see that (6a) and (6b) hold, one makes use of the formulas n I (x) = ∇dist ± (x, I) and ∂ t dist ± (x, I) = −H I · n I (P I x) valid in a neighborhood of I(t). Formula (6c) is an immediate consequence of the equality H I = −(∇ · n I )n I valid on the interface I(t) and the Lipschitz continuity of both sides of the equation.
2.2. The relative energy inequality. Our argument is based on a relative energy method. As the Modica-Mortola trick will play an important role in the definition of the relative energy, we introduce the function Given a smooth solution u ε to the Allen-Cahn equation (1) and a surface I(t) which evolves smoothly by mean curvature flow, we define the relative energy E[u ε |I] as Introducing the short-hand notation n ε := ∇u ε |∇u ε | (9a) (with n ε (x, t) ∈ S d−1 arbitrary but fixed in case |∇u ε | = 0) and writing we see that the relative energy consists of two contributions: The first term controls the local lack of equipartition of energy between the terms ε 2 |∇u ε | 2 and 1 ε W (u ε ), while the second term controls the local deviation of the normals n ε and n I . Note that the latter term also controls the distance to the interface I(t) (since |ξ| ≤ max{1 − c dist 2 (x, I), 0}). We furthermore introduce the notation motivated by the fact that H ε will play a role of a curvature vector.
The key step in our analysis is the following Gronwall-type estimate for the relative energy.

2.3.
Coercivity properties of the relative energy functional. For the proof of the Gronwall-type inequality of Theorem 2, we shall need the following coercivity properties of the relative energy.

2.4.
Time evolution of the relative energy functional. The main step in the proof of Theorem 2 is the derivation of the following formula; by estimating the right-hand side using the abovementioned coercivity properties and equations (6a)-(6c), we will derive the Gronwall-type inequality of Theorem 2.

Lemma 4. Let u ε be a solution to the Allen-Cahn equation (1) and let I = I(t) be a smooth solution to mean curvature flow. Let ξ be as defined in (4). The time evolution of the relative energy is then given by
Proof. By direct computation, we obtain With the definitions (9a) and (9b), we obtain We exploit the symmetry of the Hessian ∇ 2 ψ ε The computation (13) below then implies by adding zero Completing the squares and adding zero, we obtain (12).

Derivation of the Gronwall inequality.
Proof of Theorem 2. Using the estimates of Lemma 3 we can control the terms on the right-hand side of the identity (12). Using (6a), (6b), and the bound ||∇H I || L ∞ ≤ C(I(t)), the last four lines of (12) may be estimated by which by (10b) and (10d) is bounded by C(I(t))E[u ε |I].
The third line on the right-hand side of (12) can be estimated aŝ Thus, it only remains to estimate the second and the fourth term on the right-hand side of (12).
Concerning the second term, we use the fact that (ξ · ∇)H I ≡ 0 holds in a neighborhood of I(t), Young's inequality, and (7) to deducê Consequently, Lemma 3 implies that the fourth line on the right-hand side of (12) is bounded by CE[u ε |I].
It only remains to bound the term in the second line of the right-hand side of (12). To this aim, we complete the square and estimatê Inserting the estimates (6c) and (6d) and using the fact that H I = (H I · ξ)ξ + O(dist 2 (x, I)), we obtain By Lemma 3, we see that these terms are estimated by CE[u ε |I].