\(L_2\)-theory for two incompressible fluids separated by a free interface
type: Item
instance of: scholarly article
title: \(L_2\)-theory for two incompressible fluids separated by a free interface
zbMATH Open document ID: 1416.35182
DOI: 10.12775/TMNA.2018.019
author: Irina V. Denisova
author: V. A. Solonnikov
published in: Topological Methods in Nonlinear Analysis
publication date: 2019-02-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tmna/1534557631
review text: It is a very interesting and technical paper. The 3D non-steady flow of two viscous, incompressible, immiscible fluids in a bounded container is considered, related with the motion of a drop in a liquid. A surface tension exists on the interface between the fluids and the flow is governed by the Navier-Stokes system. Some methods introduced in [the second author, Lect. Notes Math. 1812, 123--175 (2003; Zbl 1038.35063)], [the authors, J. Math. Sci., New York 185, No. 5, 668--686 (2012; Zbl 1278.35175); translation from Zap. Nauchn. Semin. POMI 397, 20--52 (2011)], [the second author, J. Math. Sci., New York 198, No. 6, 761--827 (2014; Zbl 1309.35093); translation from Probl. Mat. Anal. 75, 93--152 (2014)] are used, giving new estimates of the solution. The problem is treated in a particular Sobolev-Slobodetskii space. A modified Hanzawa transformation is used to obtain a flow in two domains separated by a specific sphere $S_{R_0}(0)$ centered in origin. The stability of a rest state is studied, for small velocities and mass forces. The transformed problem is linearized. An exponential inequality for the corresponding energy is obtained. The results for the linearized problem and some estimates for the nonlinear terms give an exponential decay of a global solution. The authors specify that the (unsteady) interface tends to a sphere which in general is different from $S_{R_0}(0)$.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 76D45
zbMATH DE Number: 7029868
zbMATH Keywords: two-phase problems
zbMATH Keywords: Navier-Stokes system
zbMATH Keywords: surface tensions
zbMATH Keywords: unsteady interfaces
zbMATH Keywords: Sobolev-Slobodetskii spaces
reviewed by: Gelu Paşa
MaRDI profile type: Publication
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OpenAlex ID: W2888316329
description: scientific article; zbMATH DE number 7029868