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On a linearized Mullins-Sekerka/Stokes system for two-phase flows
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany |
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.
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. |
show all references
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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