On a linearized Mullins-Sekerka/Stokes system for two-phase flows

Helmut Abels; Andreas Marquardt

doi:10.3934/dcdss.2020467

Article Contents

2021, Volume 14, Issue 11: 3973-3987. Doi: 10.3934/dcdss.2020467 open access

This issue Previous Article Dimension reduction of thermistor models for large-area organic light-emitting diodes Next Article Cahn–Hilliard–Brinkman systems for tumour growth

On a linearized Mullins-Sekerka/Stokes system for two-phase flows

Helmut Abels^, and
Andreas Marquardt

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

^* Corresponding author: Helmut Abels
^* Corresponding author: Helmut Abels

Received: June 2020

Revised: September 2020

Early access: November 2020

Published: November 2021

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We study a linearized Mullins-Sekerka/Stokes system in a bounded domain with various boundary conditions. This system plays an important role to prove the convergence of a Stokes/Cahn-Hilliard system to its sharp interface limit, which is a Stokes/Mullins-Sekerka system, and to prove solvability of the latter system locally in time. We prove solvability of the linearized system in suitable $ L^2 $-Sobolev spaces with the aid of a maximal regularity result for non-autonomous abstract linear evolution equations.

Keywords:

Mathematics Subject Classification: Primary: 76T99; Secondary: 35Q30, 35Q35, 35R35, 76D05, 76D45.

Citation:

FullText(HTML)

Related Papers

Cited by

References

[1]	H. Abels and Y. Liu, Sharp interface limit for a Stokes/Allen-Cahn system, Archives for Rational Mechanics and Analysis, 229 (2018), 417-502. doi: 10.1007/s00205-018-1220-x.
[2]	H. Abels and A. Marquardt, Sharp interface limit of a Stokes/Cahn-Hilliard system, part II: Approximate solutions, preprint, arXiv: 2003.14267.
[3]	H. Abels and A. Marquardt, Sharp interface limit of a Stokes/Cahn-Hilliard system, part I: Convergence result, preprint, arXiv: 2003.03139.
[4]	H. Abels and M. Wilke, Well-posedness and qualitative behaviour of solutions for a two-phase Navier-Stokes/Mullins-Sekerka system, Interfaces and Free Boundaries, 15 (2013), 39-75. doi: 10.4171/IFB/294.
[5]	G. Alessandrini, A. Morassi and E. Rosset, The linear constraint in Poincaré and Korn type inequalities, Forum Mathematicum 20 (2006), no. 3,557–-569. doi: 10.1515/FORUM.2008.028.
[6]	N. D. Alikakos, P. W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Archive for Rational Mechanics and Analysis, 128 (1994), 165-205. doi: 10.1007/BF00375025.
[7]	W. Arendt, R. Chill, S. Fornaro and C. Poupaud, $L^p$-Maximal regularity for non-autonomous evolution equations, Journal of Differential Equations, 237 (2007), 1-26. doi: 10.1016/j.jde.2007.02.010.
[8]	X. Chen, D. Hilhorst and E. Logak, Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces and Free Boundaries, 12 (2010), 527-549. doi: 10.4171/IFB/244.
[9]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, second ed., Springer Monographs in Mathematics, 2011. doi: 10.1007/978-0-387-09620-9.
[10]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.
[11]	J. Pruess and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-27698-4.
[12]	S. Schaubeck, Sharp Interface Limits for Diffuse Interface Models, Ph.D. thesis, University of Regensburg, urn: nbn: de: bvb: 355-epub-294622, 2014.
[13]	Y. Shibata and S. Shimizu, On a resolvent estimate of the interface problem for the Stokes system in a bounded domain, Journal of Differential Equations, 191 (2003), 408-444. doi: 10.1016/S0022-0396(03)00023-8.
[14]	——, On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, Journal für die reine und angewandte Mathematik 615 (2007), 1–53.
[15]	V. A. Solonnikov and V. E. Ščadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov. 125 (1973), 196–210,235, Boundary value problems of mathematical physics, 8.