https://neckpinch.com/authors/andrea-poiatti/New arXiv papers by Andrea Poiatti, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.00732arxiv:2610.00732Wed, 30 Sep 2026 00:00:00 GMTAndrea Poiatti — 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.Mean curvature flowWeak & singular flowsLong-time existence & convergencePhase fields & multiphase flows https://arxiv.org/abs/2607.03930arxiv:2607.03930Sat, 04 Jul 2026 00:00:00 GMTTim Laux, Andrea Poiatti — We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate L² distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows https://arxiv.org/abs/2507.08783arxiv:2507.08783Fri, 11 Jul 2025 00:00:00 GMTAndrea Poiatti — In this contribution we introduce a novel weak solution concept for two-phase volume-preserving mean curvature flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions extend the ones proposed by Hensel-Laux [J. Differential Geom. 130, 209-268 (2025)] for the standard mean curvature flow, and consist in evolving varifolds coupled with the phase volumes by a transport equation. First, we show that, in the same setting as in Takasao [Arch. Ration. Mech. Anal. 247, 52 (2023)], any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to our new definition. Secondly, we crucially introduce a new notion of volume-preserving gradient-flow calibrations, allowing the extended velocity vector field to point in the normal direction on the interface. We show that any sufficiently regular strong solution is calibrated in this sense. Finally, we prove that any classical solution to volume-preserving mean curvature flow, which is then automatically a calibrated flow, is unique in the class of our new varifold solutions.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flowsStability & uniquenessLong-time existence & convergencePhase fields & multiphase flows