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

  • * Corresponding author: Helmut Abels

    * Corresponding author: Helmut Abels 
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  • 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.

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


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  • [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. AlikakosP. 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. ArendtR. ChillS. 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. ChenD. 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. McLeanStrongly 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.
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