Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T14:14:10.869Z Has data issue: false hasContentIssue false
  • English
  • Français

A moving boundary problem for the Stokes equations involving osmosis: Variational modelling and short-time well-posedness

Published online by Cambridge University Press:  24 November 2015

FRIEDRICH LIPPOTH ,
MARK A. PELETIER  and
GEORG PROKERT
FRIEDRICH LIPPOTH
Affiliation:
Institute of Applied Mathematics, Leibniz University Hannover, Welfengarten 1, D-30167 Hannover, Germany email: lippoth@ifam.uni-hannover.de
MARK A. PELETIER
Affiliation:
Faculty of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, the Netherlands emails: m.a.peletier@tue.nl, g.prokert@tue.nl
GEORG PROKERT
Affiliation:
Faculty of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, the Netherlands emails: m.a.peletier@tue.nl, g.prokert@tue.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short-time.

Keywords

variational modellingStokes equationsosmosismoving boundary problemmaximal continuous regularity
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 4 , August 2016 , pp. 647 - 666
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Amann, H. (1995) Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel, xxxvi+335 pp.CrossRefGoogle Scholar
[2] Antanovskii, L. K. (1993) Analyticity of a free boundary in plane quasi-steady flow of a liquid form subject to variable surface tension. In: Proc. of Conference: The Navier-Stokes Equations II: Theory and Numerical Methods, Oberwolfach 1991, Berlin: Springer, pp. 1–16.Google Scholar
[3] Bergner, M., Escher, J. & Lippoth, F. (2012) On the blow up scenario for a class of parabolic moving boundary problems. Nonlinear Anal.: T. M&A 75, 3951–3963.CrossRefGoogle Scholar
[4] Bonaschi, G., Carrillo, J., Di Francesco, M. & Peletier, M. Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D. submitted, arXiv: 1310.411.Google Scholar
[5] Cheng, C. H. A, Coutand, D. & Shkoller, S. (2007) Navier-Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 39, 742–800.Google Scholar
[6] Escher, J. (2004) Classical solutions for an elliptic parabolic system. Interfaces Free Boundaries 6, 175–193.Google Scholar
[7] Escher, J. & Prokert, G. (2006) Analyticity of solutions to nonlinear parabolic equations on manifolds and an application to Stokes flow. J. Math. Fluid Mech. 8, 1–35.Google Scholar
[8] Escher, J. & Simonett, G. (1997) Classical solutions for Hele-Shaw models with surface tension. Adv. Differ. Equ. 2, 619–642.Google Scholar
[9] Frischmuth, K. & Hänler, M. (1999) Numerical analysis of the closed osmometer problem. Z. Angew. Math. Mech. 79, 107–116.3.0.CO;2-E>CrossRefGoogle Scholar
[10] Günther, M. & Prokert, G. (1997) Existence results for the quasistationary motion of a capillary liquid drop. Z. Anal. ihre Anwendungen 16, 311–348.Google Scholar
[11] Hopper, R.W. (1990) Plane stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349–375.Google Scholar
[12] Kneisel, C. (2008) Über das Stefan-Problem mit Oberflächenspannung und thermischer Unterkühlung. PhD Thesis, Universität Hannover 2007; VDM Verlag Dr. Müller.Google Scholar
[13] Lippoth, F. & Prokert, G. (2012) Classical solutions for a one phase osmosis model. J. Evol. Equ. 12 (2), 413–434.Google Scholar
[14] Lippoth, F. & Prokert, G. (2014) Stability of equilibria for a two-phase osmosis model. NoDEA Nonlinear Differ. Equ. Appl. 21, 129–149.Google Scholar
[15] Lunardi, A. (1989) Maximal space regularity in inhomogeneous initial boundary value parabolic problems. Num. Funct. Anal. Opt. 10 (3 and 4), 323–349.Google Scholar
[16] Lunardi, A. (1995) Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, xviii+424 pp.Google Scholar
[17] Meurs, P. J. P. v. (2011) Osmotic Cell Swelling in the Fast Diffusion Limit. MSc Thesis, Eindhoven University of Technology, http://alexandria.tue.nl/extra1/afstversl/wsk-i/meurs2011.pdf Google Scholar
[18] Mielke, A. (2005) Evolution in rate-independent systems. In: Handbook of Differential Equations: Evolutionary Differential Equations, Amsterdam: North-Holland, pp. 461–559.Google Scholar
[19] Peletier, M. A. (2014) Variational Modelling: Energies, Gradient Flows, and Large Deviations, arxiv:1402.1990.Google Scholar
[20] Rayleigh, L. (1913) On the motion of a viscous fluid. London, Edinburgh Dublin Phil. Mag. J. Sci. 26, 776–786.Google Scholar
[21] Rubinstein, L. & Martuzans, B. (1995) Free Boundary Problems Related to Osmotic Mass Transfer Through Semipermeable Membranes, Gakkotosho, Tokyo, vi+205 pp.Google Scholar
[22] Solonnikov, V. A. (1999) On quasistationary approximation in the problem of motion of a capillary drop. In: Escher, J. & Simonett, G. (editors), Topics in Nonlinear Analysis, Birkhäuser, Basel, Progress in Nonlinear Differential Equations and their Applications 35, pp. 643–671.Google Scholar
[23] Vainberg, M. M. & Trenogin, V. A. (1974) Theory of Branching of Solutions of Non-linear Equations, Noordhoff, Leyden, xxvi+485 pp.Google Scholar
[24] Zaal, M. M. (2008) Linear Stability of Osmotic Cell Swelling. MSc Thesis, Vrije Universiteit Amsterdam. http://www.few.vu.nl/~mzl400/bin/scriptie.pdf Google Scholar
[25] Zaal, M. M. (2012) Cell swelling by osmosis: A variational approach. Interfaces Free Boundaries 14, 487–520.Google Scholar
[26] Zaal, M. M. (2013) Variational Modeling of Parabolic Free Boundary Problems. PhD Thesis, Vrije Universiteit Amsterdam. http://dare.ubvu.vu.nl/bitstream/1871/40209/1/dissertation.pdf Google Scholar