Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

2025/05/29 by Andrea Chiesa, Chiesa, Andrea, Keisuke Takasao +1 · 1 citation
Mathematics · #35K93 #53E10 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35K93 #msc:53E10

paper · pdf · doi:10.48550/arxiv.2505.23222

arxiv created 2026/07/30 · arxiv updated 2026/08/03

Abstract

In this paper, we propose a new notion of Brakke inequality for volume preserving mean curvature flow. We show the existence of integral varifolds solving the flow globally-in-time in the corresponding Brakke sense using the phase field method. Moreover, such varifolds are solutions to volume preserving mean curvature flow in the L²-flow sense as well. We thus extend a previous result by one of the authors [25].

Citations

• The space-time-Grassmann measure of the Brakke flow
• Asymptotic behavior of a diffused interface volume-preserving mean curvature flow
• Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications
• Weak-strong uniqueness for volume-preserving mean curvature flow
• The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions
• Volume preserving mean curvature flow for star-shaped sets
• Convergence of the Allen-Cahn Equation to multi-phase mean-curvature\n flow
• Convergence of the thresholding scheme for multi-phase mean-curvature\n flow
• Convergence of thresholding schemes incorporating bulk effects
• Global solutions to the volume-preserving mean-curvature flow
• The Allen-Cahn Action functional in higher dimensions

Cited by

• Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

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• Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow