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

TL;DR

This paper introduces a new concept of weak solutions for a coupled Navier-Stokes and Mullins-Sekerka model describing two incompressible, viscous fluids with different densities, proving their global existence and consistency with classical solutions.

Contribution

It proposes a novel weak solution framework that handles different fluid densities and incorporates energy dissipation and boundary contact conditions.

Findings

Established global existence of weak solutions.

Proved consistency with classical solutions.

Included energy dissipation and boundary contact conditions.

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 Gr\"{u}n (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 \`a 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 R\"{o}ger (Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 26, 2009).