Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system
Abstract
The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.
Related
- Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
- Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy
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- Global finite-energy weak solutions to the Navier--Stokes--Maxwell in 1D
- Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law
BibTeX
@misc{poiatti2026global,
title = {{Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system}},
author = {Andrea Poiatti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00732},
eprint = {2610.00732},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00732},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system
AU - Andrea Poiatti
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00732
UR - https://arxiv.org/abs/2610.00732
AB - The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.
ER -