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Weak Solutions to a Sharp Interface Model for a Two-Phase Flow of Incompressible Viscous Fluids with Different Densities

Helmut Abels, Andrea Poiatti

TL;DR

The paper establishes global existence of weak (varifold) solutions for a sharp-interface two-phase flow with density and viscosity contrasts, uniting Navier–Stokes motion with Mullins–Sekerka interface dynamics. It introduces a De Giorgi-type energy-dissipation framework and a weak contact-angle formulation, achieving a sharp energy inequality in the presence of advection by a divergence-free velocity field. The authors construct a time-discrete minimizing-movement scheme, perform careful compactness arguments, and pass to the limit to obtain a varifold NS–MS solution with a generalized mean curvature and Gibbs–Thomson law, along with consistency results tying weak solutions to classical ones. This provides a robust variational structure for sharp-interface two-phase flows with unmatched densities and sets the stage for potential weak-strong uniqueness results and further PDE analysis in Capillarity-driven flows.

Abstract

In this paper we consider the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier-Stokes and Mullins-Sekerka type parts, and can be obtained from the sharp interface limit of the diffuse interface model proposed by the first author, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012). We introduce a new notion of weak solutions and prove its global in time existence, together with a consistency result of smooth weak solutions with the classical Navier-Stokes-Mullins-Sekerka system. Our new notion of solution allows to include the case of different densities of the two fluids, a sharp energy dissipation principle à la De Giorgi, together with a weak formulation of the constant contact angle condition at the boundary, which were left open in the previous notion of solution proposed by the first author and Röger (Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2009).

Weak Solutions to a Sharp Interface Model for a Two-Phase Flow of Incompressible Viscous Fluids with Different Densities

TL;DR

The paper establishes global existence of weak (varifold) solutions for a sharp-interface two-phase flow with density and viscosity contrasts, uniting Navier–Stokes motion with Mullins–Sekerka interface dynamics. It introduces a De Giorgi-type energy-dissipation framework and a weak contact-angle formulation, achieving a sharp energy inequality in the presence of advection by a divergence-free velocity field. The authors construct a time-discrete minimizing-movement scheme, perform careful compactness arguments, and pass to the limit to obtain a varifold NS–MS solution with a generalized mean curvature and Gibbs–Thomson law, along with consistency results tying weak solutions to classical ones. This provides a robust variational structure for sharp-interface two-phase flows with unmatched densities and sets the stage for potential weak-strong uniqueness results and further PDE analysis in Capillarity-driven flows.

Abstract

In this paper we consider the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier-Stokes and Mullins-Sekerka type parts, and can be obtained from the sharp interface limit of the diffuse interface model proposed by the first author, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012). We introduce a new notion of weak solutions and prove its global in time existence, together with a consistency result of smooth weak solutions with the classical Navier-Stokes-Mullins-Sekerka system. Our new notion of solution allows to include the case of different densities of the two fluids, a sharp energy dissipation principle à la De Giorgi, together with a weak formulation of the constant contact angle condition at the boundary, which were left open in the previous notion of solution proposed by the first author and Röger (Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2009).
Paper Structure (42 sections, 9 theorems, 360 equations)

This paper contains 42 sections, 9 theorems, 360 equations.

Table of Contents

  1. Introduction
  2. Functional setting and assumptions
  3. Measures and varifolds
  4. Main assumptions
  5. Main results
  6. Technical Lemmas
  7. Preliminary results
  8. Flow map and related properties
  9. Proof of Theorem \ref{['thm1']}
  10. Approximating velocity
  11. First discretized step
  12. Minimizing movement step
  13. Energy estimates for the advective Mullins-Sekerka flow
  14. Approximating scheme for the fluid equation: interval $[0,h]$
  15. Iterative scheme at step $n>1$
  16. ...and 27 more sections

Key Result

Theorem 3.8

Let $d \in \{2,3\}$, $\Omega\subset \mathbb R^d$ be a bounded domain with smooth boundary, and $\rho,\nu$ as in assumptions (M0)-(M1). Let $\mathbf{v}_0\in \mathbf L^2_\sigma(\Omega)$, $m_0 \in (0,\mathcal{L}^d( \Omega))$, $\chi_0 \in \mathcal{M}_{m_0}$, $c_0\in (0,\infty)$, $\gamma \in (0, \pi/2]$. for almost every $t\in(0,{T^*})$.

Theorems & Definitions (24)

  • Definition 3.1: Admissible couples of evolving phase indicators and varifolds
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4: Varifold solutions to Navier-Stokes-Mullins-Sekerka system
  • Remark 3.5: Total energy dissipation inequality
  • Remark 3.6
  • Remark 3.7: Kinetic and curvature potentials
  • Theorem 3.8: Existence of global varifold solutions to the Navier-Stokes-Mullins-Sekerka system
  • Theorem 3.9: Consistency
  • Remark 3.10
  • ...and 14 more