# American Institute of Mathematical Sciences

November  2021, 14(11): 3973-3987. doi: 10.3934/dcdss.2020467

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

 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

* Corresponding author: Helmut Abels

Received  June 2020 Revised  September 2020 Published  November 2021 Early access  November 2020

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.

Citation: Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 3973-3987. doi: 10.3934/dcdss.2020467
##### 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.

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##### 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.
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