Canonical BV–Brakke Flows

• Canonical BV–Brakke flows are weak solutions to mean curvature flow that use BV functions to capture evolving finite-perimeter sets and singular interfaces.
• They are constructed through variational methods such as Allen–Cahn, elliptic regularization, and time-discrete algorithms while satisfying a Brakke inequality and BV transport identity.
• Robust epsilon-regularity results and a comprehensive regularity theory enable controlled analysis of singularity formation and ensure the uniqueness and stability of these flows.

Canonical $\mathrm{BV}$-Brakke flows represent a class of weak solutions to mean curvature flow (MCF) where evolving hypersurfaces or networks are described not by smooth immersions but by families of Radon measures, varifolds, or open partitions whose boundaries have bounded variation (BV). This notion encompasses and refines Brakke’s original measure-theoretic formulation of mean curvature flow, incorporating the structure of evolving finite-perimeter sets and enabling the handling of interfaces with singularities such as triple junctions. Canonical BV–Brakke flows arise, for instance, as the limiting dynamics of phase-field models (e.g., Allen–Cahn), elliptic/parabolic regularization, or discrete algorithmic approximations, and are equipped with both a Brakke inequality for the interface measure and a BV transport identity for the phases. These properties provide a robust framework for addressing existence, uniqueness (in suitable regimes), regularity, and the description of singularity formation in mean curvature-type geometric evolution problems.

1. Foundational Concepts and Definitions

Canonical $\mathrm{BV}$-Brakke flows are constructed to resolve the weaknesses and nonuniqueness seen in general Brakke flows by anchoring the geometric evolution to BV functions—functions of bounded variation that naturally describe finite-perimeter sets and their moving boundaries. The central objects are:

• Varifolds: Time-indexed families $\{V_t\}_{t \geq 0}$ of integral varifolds representing generalized surfaces, satisfying an energy dissipation (Brakke) inequality.
• BV Structure (Phases/Partition): Distinct open sets $\{E_i(t)\}_{i=1}^N$ partitioning space over time, each with characteristic functions $\chi_{E_i(t)} \in BV$ and reduced boundaries $\partial^* E_i(t)$ that encode the evolving interface.
• Generalized Mean Curvature Vector: For almost every $t$, there exists $h(x, V_t)\in L^2_\mathrm{loc}$ with

$$\delta V_t(g) = -\int g(x) \cdot h(x,V_t) \, d\|V_t\|(x)$$

for each test vector field $g$.

The measure-theoretic evolution is encoded by the Brakke inequality, and (crucially for canonical flows) the BV transport formula links changes in each phase’s volume to the flux of mean curvature across phase boundaries: $$\frac{d}{dt} \int_{E_i(t)} \phi(x,t) dx = \int_{\partial^* E_i(t)} \phi(x,t) [h(x,V_t) \cdot \nu_i(x)] d\mathcal{H}^n + \int_{E_i(t)} \partial_t \phi(x,t) dx,$$ for all test functions $\phi$, see (Stuvard et al., 2021).

2. Variational and Approximation Frameworks

Canonical BV–Brakke flows are often constructed as limits of variational regularization or time-discrete approximation schemes:

• Allen–Cahn Approximation: The diffuse interface energy

$$e_\varepsilon(x,t) = \frac{\varepsilon}{2}|\nabla u_\varepsilon(x,t)|^2 + \frac{1}{\varepsilon} F(u_\varepsilon(x,t))$$

is associated to measures $d\mu_\varepsilon(x,t)=e_\varepsilon(x,t)dV(x)$, which converge (as $\varepsilon \to 0$) to rectifiable limiting measures $p_t$ satisfying a Brakke-type inequality (Pisante et al., 2013, Nguyen et al., 2020, Tashiro, 2023).

• Elliptic Regularization: Approximates sharp interface evolution by smoothing, with the limit interface measure $\mu$ identified with the BV boundary measure $|\nabla' \chi_E|$ and tangent planes $T_{(x,t)} \mu = T_{(x,t)} (\partial^* E)$ almost everywhere. The BV transport formula holds exactly in the limit (Tashiro, 2023, Tashiro, 2023).
• Time-Discrete Partition Algorithms: Iterative schemes based on area-decreasing Lipschitz deformations and mollified mean curvature motion (for clusters/partitions), with existence and regularity established for any closed rectifiable initial interface (Stuvard et al., 2019, Stuvard et al., 2021, Stuvard, 2023, Buet et al., 1 Oct 2025, Sagueni, 8 Sep 2025).
• Level Set and Viscosity Methods: For almost every level set of the viscosity solution, the indicator function of the super-level set evolves as a BV solution saturating the optimal energy dissipation rate (Ullrich et al., 2023).

3. Regularity Theory and Epsilon-Regularity Results

A robust regularity theory has been developed for canonical BV–Brakke flows, leveraging structural features and geometric monotonicity:

• Local Regularity up to the End-Time: If at a space–time point the Gaussian density is near one, and the flow is close (in measure) to a plane, then the support of the flow is a smooth mean curvature flow up to that time, extending White’s local regularity theorem to arbitrary Brakke flows (Stuvard et al., 2022, Stuvard, 2023, Lahiri, 2016).
• $\varepsilon$-Regularity near Triple Junctions: For BV–Brakke flows with multi-phase (cluster-like) structure, if the flow is L²-close to a static multiplicity-one triple junction, then locally the flow consists of three smooth sheets meeting at a $C^{1,\alpha}$ junction, matching the expected 120° angle geometry (Tonegawa et al., 2015, Stuvard et al., 3 Oct 2025). The critical structural assumption required for the regularity (concerning the behavior of 1-dimensional slices) is always satisfied for canonical BV–Brakke flows arising from clusters or mod 3 currents.
• Viscosity and Parabolic Techniques: Regularity proofs using viscosity solutions and iteration of flatness improvement, allowing $C^{1,\alpha}$ boundary regularity for flows close to a half-plane and up to the fixed boundary (Gasparetto, 2022).

Table: Epsilon-Regularity Results

Setting	Regularity Conclusion	Source
Gaussian density near 1 (end-time)	$C^\infty$ graph up to final time	(Stuvard et al., 2022)
Cluster-like flow, near triple junction	Three $C^{1,\alpha}$ sheets at $120^\circ$	(Stuvard et al., 3 Oct 2025)
Planar network flows, near triple junction	$C^{1,5}$ regular triple junction structure	(Tonegawa et al., 2015)
Near stationary half-plane (with boundary)	$C^{1,\alpha}$ graphical regularity	(Gasparetto, 2022)

These results provide unconditional regularity for canonical flows provided certain density and structural smallness are satisfied.

4. Existence and Selection Mechanisms

Global-in-time existence of canonical BV–Brakke flows has been established for broad classes of initial data:

• Partition Flows/Clusters: For any open partition of space with finite interface measure, there exists a global BV–Brakke flow, and for almost all times, the interfaces are unions of smooth (e.g., $W^{2,2}$) curves (in planar cases) or higher-dimensional smooth sheets (Kim et al., 2020, Stuvard et al., 2021, Stuvard, 2023).
• With Fixed Boundary Conditions: BV–partition flows with prescribed boundary (in strictly convex domains) globally exist and remain compatible with the topological constraint, and the long-time limit yields a stationary varifold spanning the boundary (dynamical solution to Plateau’s problem) (Stuvard et al., 2019, Stuvard, 2023).
• Multi-Phase Flows and Uniqueness: The BV structure enforces volume–change identities for grains, anchoring the evolution to the partition and resolving nonuniqueness in regimes where the density ratio is suitably controlled (Stuvard et al., 2021, Ullrich et al., 2023).

5. Analytical Structure and Equivalence of Definitions

The weak solution framework for BV–Brakke flows is robust under multiple definitions:

• Equivalence of Brakke Formulations: Pointwise, strong (allowing time-dependent test functions), and integrated (weak) versions of the Brakke inequality are equivalent if the Brakke variation is upper-semicontinuous and the translation term is continuous (Lahiri, 2017). The corrected proof ensures full rigor for time-dependent test functions and removes ambiguity for flows in BV.
• Canonical Structure of Measures and Tangent Planes: The space–time measure induced by the BV flow and Brakke flow coincide on the reduced boundary, and the approximate tangent space for the measure is that of the moving interface almost everywhere (Tashiro, 2023, Tashiro, 2023).
• Gradient Flow and Energy Dissipation: For almost every level set of the viscosity solution, the canonical BV–Brakke flow saturates the optimal energy dissipation rate, confirming the gradient flow (curve of maximal slope) structure (Ullrich et al., 2023).

6. Singularities, Stratification, and Avoidance

Canonical BV–Brakke flows feature refined control over singularity sets and robust geometric properties:

• Singular Set Dimension: In one-dimensional network flows, the space–time singular set has parabolic Hausdorff dimension at most 1, with the set of “bad” times of ordinary Hausdorff dimension at most $1/2$ (Tonegawa et al., 2015).
• Stratification: The finest structure of singularities, via White’s stratification theorem, shows that except for a closed set of low dimension, the flow is “classical”—away from the singular set, the canonical BV–Brakke flow is smooth or consists of triple junctions with the correct angles.
• Avoidance Principle and Mass Non-Vanishing: With initial data comprising boundaries of finite–perimeter domains, canonical spacetime BV–Brakke flows are nontrivial as long as the initial mass is nonzero. Furthermore, the support of the mass measure does not cross barriers given by evolving smooth MCF hypersurfaces, and in the codimension 1 setting, coincides with the smooth MCF as long as it exists (Sagueni, 8 Sep 2025).

7. Extensions: Forcing, Volume-Preserving, and General Data

The canonical BV–Brakke flow paradigm extends to various geometric and analytic settings:

• Volume-Preserving Brakke Flows: A modified Brakke inequality accommodates evolution with a volume constraint, including an error term compatible with approximation schemes via the Allen–Cahn equation with volume penalization. Integral varifolds are constructed as solutions in this sense (Chiesa et al., 29 May 2025).
• General Varifolds and Approximate Flows: The approximate mean curvature flow for general varifold data, applicable beyond smooth or rectifiable sets (including point clouds), is constructed via iterated push-forwards and canonically coupled to time. Under rectifiability, the limit yields a spacetime Brakke flow (Buet et al., 1 Oct 2025).
• Gap Theorems and Rigidity: In the Allen–Cahn context, a gap theorem excludes non-flat eternal solutions near unit density, implying that, in the mean convex case, all diffuse energy concentrates on the canonical interface, and there is no cancellation in the BV convergence (Nguyen et al., 2020).

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Canonical BV–Brakke flows thus provide a comprehensive theory for geometric evolution by mean curvature in the presence of singularities, capturing both the dissipative (Brakke) and geometric (BV) aspects of evolving interfaces or networks. The framework ensures existence, stability, regularity near regular and singular points (even in the presence of triple junctions), and uniquely identifies the solution through BV–volume identities under natural conditions. It is robust with respect to numerically and variationally motivated approximations, can treat fixed boundary and multi-phase scenarios, and is flexible enough to address further constraints (e.g., volume preservation), making it the central object underpinning modern measure-theoretic mean curvature flow.