Paper · May 30, 2017

Well-Posedness of a Navier-Stokes/Mean Curvature Flow system

Helmut Abels, Maximilian Moser

math.APWider flowsarXiv:1705.10832
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Abstract

We consider a two-phase flow of two incompressible, viscous and immiscible fluids which are separated by a sharp interface in the case of a simple phase transition. In this model the interface is no longer material and its evolution is governed by a convective mean curvature flow equation, which is coupled to a two-phase Navier-Stokes equation with Young-Laplace law. The problem arises as a sharp interface limit of a diffuse interface model, which consists of a Navier-Stokes system coupled with an Allen-Cahn equation. We prove existence of strong solutions for sufficiently small times and regular initial data.

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Submitted
May 30, 2017
Subjects
math.AP

Comments: 32 pages

Mean curvature flowPhase fields & multiphase flows

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Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Convergence of the Allen-Cahn Equation to the Mean Curvature Flow with 90^∘-Contact Angle in 2D (Helmut Abels and Maximilian Moser, 2018)3Abels 2018 Convergence of the Scalar- and Vector-Valued Allen-Cahn Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions (Maximilian Moser, 2021)5Moser 2021 Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to 90° (Helmut Abels and Maximilian Moser, 2021)4Abels 2021 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)7Fischer 2022 Stability of multiphase mean curvature flow beyond circular topology changes (Julian Fischer, Sebastian Hensel, Alice Marveggio et al., 2024)6Fischer 2024 Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow (Helmut Abels, Christoph Hurm and Maximilian Moser, 2024)2Abels 2024 Generalised stochastic curvature flow in d ≥ 2, and sharp interface limit for the stochastic Allen-Cahn equation with nonlinear diffusion (Weijun Xu and Shuhan Zhou, 2026)8Xu 2026 Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system (Andrea Poiatti, 2026)1Poiatti 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system

    Andrea PoiattiWider flows

    Topics Phase fields & multiphase flows · Shared terms “Navier Stokes”, “Stokes Mean”, “Flow system”

  2. Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow

    Helmut Abels, Christoph Hurm and Maximilian MoserWider flows

    Topics Phase fields & multiphase flows · Same authors Helmut Abels, Maximilian Moser · Shared terms “Allen Cahn”, “Cahn equation”, “interface”

  3. Convergence of the Allen-Cahn Equation to the Mean Curvature Flow with -Contact Angle in 2D

    Helmut Abels and Maximilian MoserWider flows

    Topics Phase fields & multiphase flows · Same authors Helmut Abels, Maximilian Moser · Shared terms “interface”, “Allen Cahn”, “Cahn equation”

  4. Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to °

    Helmut Abels and Maximilian MoserWider flows

    Topics Phase fields & multiphase flows · Same authors Helmut Abels, Maximilian Moser · Shared terms “Allen Cahn”, “Cahn equation”, “interface”

  5. Convergence of the Scalar- and Vector-Valued Allen-Cahn Equation to Mean Curvature Flow with °-Contact Angle in Higher Dimensions

    Maximilian MoserWider flows

    Topics Phase fields & multiphase flows · Same author Maximilian Moser · Shared terms “Allen Cahn”, “Cahn equation”, “interface”

  6. Stability of multiphase mean curvature flow beyond circular topology changes

    Julian Fischer, Sebastian Hensel, Alice Marveggio et al.Wider flows

    Topics Phase fields & multiphase flows · Same author Maximilian Moser · Shared terms “interface”, “initial data”

  7. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

    Julian Fischer and Alice MarveggioWider flows

    Topics Phase fields & multiphase flows · Shared terms “Allen Cahn”, “Cahn equation”, “interface”

  8. Generalised stochastic curvature flow in , and sharp interface limit for the stochastic Allen-Cahn equation with nonlinear diffusion

    Weijun Xu and Shuhan ZhouWider flows

    Topics Phase fields & multiphase flows · Shared terms “sharp interface”, “interface limit”, “Allen Cahn”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.