Journal of Evolution Equations

, Volume 12, Issue 2, pp 413–434 | Cite as

Classical solutions for a one-phase osmosis model

Article
First Online:

Abstract

For a moving boundary problem modeling the motion of a semipermeable membrane by osmotic pressure and surface tension, we prove the existence and uniqueness of classical solutions on small time intervals. Moreover, we construct solutions existing on arbitrary long time intervals, provided the initial geometry is close to an equilibrium. In both cases, our method relies on maximal regularity results for parabolic systems with inhomogeneous boundary data.

Mathematics Subject Classification

35R37 35K55 

Keywords

Moving boundary problem Maximal continuous regularity 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amann, H.: Multiplication in Sobolev and Besov spaces. Nonlinear analysis: a tribute in honour of Giovanni Prodi. Edited by A. Ambrosetti and A. Marino. Scuola Normale Superiore di Pisa. Quaderni. Pisa, 1991Google Scholar
  2. 2.
    Bergner, M., Escher, J., Lippoth, F.: On the blow up scenario for a class of parabolic moving boundary problems. Nonlinear Analysis T,M&A, acceptedGoogle Scholar
  3. 3.
    Denk R., Prüß J., Zacher R.: Maximal L p-regularity of parabolic problems with boundary dynamics of relaxation type. Journ. Funct. Anal. 255, 149–3187 (2008)CrossRefGoogle Scholar
  4. 4.
    Escher J., Simonett G., Prüss J.: Analytic solutions for a Stefan problem with Gibbs-Thomson correction. J. r. a. Math. 563, 1–52 (2003)MATHCrossRefGoogle Scholar
  5. 5.
    Escher J., Simonett G.: Classical solutions for Hele-Shaw models with surface tension. Adv. Differential equations 2, 619–642 (1997)MathSciNetMATHGoogle Scholar
  6. 6.
    Escher J., Simonett G.: Classical solutions of multidimensional Hele-Shaw models. SIAM Journal on Mathematical Analysis Vol. 28(Is. 5),–10281047 (1997)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Escher J.: Classical solutions for an elliptic parabolic system. Interfaces and free boundaries 6, 175–193 (2004)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kneisel, C.: \"Uber das Stefan-Problem mit Oberflächenspannung und thermischer Unterkühlung. PhD thesis, Universität Hannover, 2007; VDM Verlag Dr. Müller, 2008Google Scholar
  9. 9.
    Lunardi, A.: Maximal space regularity in inhomogeneous initial boundary value parabolic problems. Num. Funct. Anal. and Opt., 1532–2467, Vol. 10, Is. 3 & 4, 1989, 323–349Google Scholar
  10. 10.
    Martuzans B., Rubinstein, L.: Free Boundary Problems Related to Osmotic Mass Transfer Through Semipermeable Membranes. Gakkotosho, Tokyo 1995Google Scholar
  11. 11.
    Pickard,W.F.: Modelling the Swelling Assay for Aquaporin Expression. J. Math. Biol. 57 (2008), 883–903MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Verkman, A.S.: Water Channels in Cell Membranes. Annu. Rev. Physiol. 54 (1992), 97–108CrossRefGoogle Scholar
  13. 13.
    Verkman, A.S.: Solute and Macromolecular Diffusion in Cellular Aqueous Compartments. Trends Biochem. Sci. 23 (2000), 27–33Google Scholar
  14. 14.
    Zaal, M.: Linear Stability of Osmotic Cell Swelling. MSc thesis, Vrije Universiteit Amsterdam, 2008Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Institute of Applied MathematicsLeibniz University of HanoverHannoverGermany
  2. 2.Faculty of Mathematics and Computer Science, TU EindhovenEindhovenThe Netherlands

Personalised recommendations