Skip to main content
SpringerLink
Log in

Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type Two-Phase Flows

  • Chapter
Recent Developments of Mathematical Fluid Mechanics

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

  • 1599 Accesses

Abstract

We first review results about existence of generalized or weak solutions for Newtonian and power-law type two-phase flows. Then we state a recent result by the authors about existence of weak solutions for diffuse interface model of power-law type two-phase flows and give a sketch of its proof. The latter part is a summary of Abels et al. (Nonlinear Anal Real World Appl 15:149–157, 2014).

Dedicated to Professor Yoshihiro Shibata’s 60th birthday

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound. 9(1), 31–65 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194(2), 463–506 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Abels, M. Röger, Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2403–2424 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Abels, M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67, 3176–3193 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Abels, L. Diening, Y. Terasawa, Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows. Nonlinear Anal. Real World Appl. 15, 149–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Bosia, Analysis of a Cahn-Hilliard-Ladyzhenskaya system with singular potential. J. Math. Anal. Appl. 397(1), 307–321 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20(2), 175–212 (1999)

    MathSciNet  MATH  Google Scholar 

  8. D. Breit, L. Diening, M. Fuchs, Solenoidal Lipschitz truncation and applications in fluid. J. Differ. Equ. 253(6), 1910–1942 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Breit, L. Diening, S. Schwarzacher, Solenoidal Lipschitz truncation for parabolic PDE’s. M3AS 23(14), 2671–2700 (2014)

    Google Scholar 

  10. M. Bulíček, F. Ettwein, P. Kaplický, D. Pražák, Dimension of the attractor for 3D flow of non-Newtonian fluid. Commun. Pure Appl. Anal. 8(5), 1503–1520 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Bulíček, P. Gwiazda, J. Málek, A. Świerczevska-Gwiazda, On Unsteady Flows of Implicitly Constituted Incompressible Fluids. SIAM J. Math. Anal. 44(4), 2756–2801 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differ. Geom. 44, 262–311 (1996)

    MathSciNet  MATH  Google Scholar 

  13. A. Debussche, L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 24(10), 1491–1514 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. I.V. Denisova, A priori estimates for the solution of the linear non stationary problem connected with the motion of a drop in a liquid medium. (Russian) Trudy Mat. Inst. Steklov 188, 3–21 (1990). [translation in Proc. Steklov Inst. Math. 3, 1–24 (1991)]

    Google Scholar 

  15. I.V. Denisova, Problem of the motion of two viscous incompressible fluids separated by a closed free interface. Mathematical problems for the Navier-Stokes equations (Centro, 1993). Acta Appl. Math. 37, 31–40 (1994)

    Google Scholar 

  16. I.V. Denisova, V.A. Solonnikov, Classical solvability of the problem of the motion of two viscous incompressible fluids. (Russian) Algebra i Analiz. 7(5), 101–142 (1995). [translation in St. Petersburg Math. J. 7(5), 755–786 (1996)]

    Google Scholar 

  17. L. Diening, M. R\(\stackrel{\circ }{\mathrm{u}}\)žička, J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Annali della Scuola Normale Superiore di Pisa Classe di scienze (5) 9(1), 1–46 (2010)

    Google Scholar 

  18. C.M. Elliott, S. Luckhaus, A generalized equation for phase separation of a multi-component mixture with interfacial free energy. preprint SFB 256 Bonn No. 195, 1991

    Google Scholar 

  19. C.M. Elliott, S. Zheng, On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Grasselli, D. Pražák, Longtime behavior of a diffuse interface model for binary fluid mixtures with shear dependent viscosity. Interfaces Free Bound. 13(4), 507–530 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. M.E. Gurtin, D. Polignone, J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(6), 815–831 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. P.C. Hohenberg, B.I. Halperin, Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977)

    Article  Google Scholar 

  23. N. Kim, L. Consiglieri, J.F. Rodrigues, On non-Newtonian incompressible fluids with phase transitions. Math. Methods Appl. Sci. 29(13), 1523–1541 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Kinnunen, J.L. Lewis, Very weak solutions of parabolic systems of p-Laplacian type. Ark. Mat. 40(1), 105–132 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Köhne, J. Prüss, M. Wilke, Qualitative behaviour of solutions for the two-phase Navier-Stokes equations with surface tension. Math. Ann. 356(2), 737–792 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. O.A. Ladyzhenskaya, Sur de nouvelles équation dans la dynamique de fluides visqueux et leur resolution globale. Troudi Mat. Inst. Stekloff CII, 85–104 (1967)

    Google Scholar 

  27. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires. (Dunod, France, 1969)

    Google Scholar 

  28. J. Málek, K.R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Handbook Differential Equation: Evolutionary equations, vol. II (Elsevier/NorthHolland, Amsterdam, 2005), pp. 371–459

    MATH  Google Scholar 

  29. J. Málek, J. Nečas, M. R\(\stackrel{\circ }{\mathrm{u}}\)žička, On the non-Newtonian incompressible fluids. M3AS 3(1), 35–63 (1993)

    Google Scholar 

  30. B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of a class of pattern formation equations. Commun. Partial Differ. Equ. 14(2), 245–297 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  31. P.I. Plotnikov, Generalized solutions to a free boundary problem of motion of a non-Newetonian fluid. Siberian Math. J. 34(4), 704–716 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Prüss, G. Simonett, On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound. 12(3), 311–34 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Prüss, G. Simonett, R. Zacher, Qualitative behavior of incompressible two-phase flows with phase transition: the case of equal densities. Interfaces Free Bound. 15(4), 405–428 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Prüss, S. Shimizu, M. Wilke, Qualitative behavior of incompressible two-phase flows with phase transitions: the case of non-equal densities. Commun. Partial Differ. Equ. 39(7), 1236–1283 (2014)

    Article  MATH  Google Scholar 

  35. P. Rybka, K-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equations. Commun. Partial Differ. Equ. 24(5–6), 1055–1077 (1999)

    Google Scholar 

  36. V.N. Starovoĭtov, On the motion of a two-component fluid in the presence of capillary forces. Mat. Zametki 62(2), 293–305 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9(1), 104–138 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the SPP 1506 “Transport Processes at Fluidic Interfaces” of the German Science Foundation (DFG) through the grant AB 285/4-1. Moreover, T. was supported by JSPS Research Fellowships for Young Scientists and by FMSP, a JSPS Program for Leading Graduate Schools in the University of Tokyo. The supports are gratefully acknowledged.

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Helmut Abels

  2. Mathematisches Institut, LMU München, 80333, München, Germany

    Lars Diening

  3. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Yutaka Terasawa

Authors
  1. Helmut Abels
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lars Diening
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yutaka Terasawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yutaka Terasawa .

Editor information

Editors and Affiliations

  1. Universität of Zürich, Institut für Mathematik, Zürich, Switzerland

    Herbert Amann

  2. University of Tokyo, Graduate School of Mathematical Sciences, Tokyo, Japan

    Yoshikazu Giga

  3. Department of Mathematics, Waseda University, Tokyo, Japan

    Hideo Kozono

  4. Research Institute for Mathematical Sciences, Kyoto University Kitashirakawa, Kyoto, Japan

    Hisashi Okamoto

  5. Department of Mathematics, Waseda University, Tokyo, Japan

    Masao Yamazaki

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Basel

About this chapter

Cite this chapter

Abels, H., Diening, L., Terasawa, Y. (2016). Existence of Weak Solutions for a Diffuse Interface Model of Power-Law Type Two-Phase Flows. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0939-9_2

Download citation

Publish with us

Policies and ethics

Search

Navigation