Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids

Helmut Abels; Matthias Röger

Annales de l'I.H.P. Analyse non linéaire (2009)

  • Volume: 26, Issue: 6, page 2403-2424
  • ISSN: 0294-1449

How to cite

top

Abels, Helmut, and Röger, Matthias. "Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids." Annales de l'I.H.P. Analyse non linéaire 26.6 (2009): 2403-2424. <http://eudml.org/doc/78940>.

@article{Abels2009,
author = {Abels, Helmut, Röger, Matthias},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {two-phase flow; Navier-Stokes equations; free boundary problems; Mullins-Sekerka system},
language = {eng},
number = {6},
pages = {2403-2424},
publisher = {Elsevier},
title = {Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids},
url = {http://eudml.org/doc/78940},
volume = {26},
year = {2009},
}

TY - JOUR
AU - Abels, Helmut
AU - Röger, Matthias
TI - Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 6
SP - 2403
EP - 2424
LA - eng
KW - two-phase flow; Navier-Stokes equations; free boundary problems; Mullins-Sekerka system
UR - http://eudml.org/doc/78940
ER -

References

top
  1. [1] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Rat. Mech. Anal., doi:10.1007/s00205-008-0160-2. Zbl1254.76158MR2563636
  2. [2] Abels H., On generalized solutions of two-phase flows for viscous incompressible fluids, Interfaces Free Bound.9 (2007) 31-65. Zbl1124.35060MR2317298
  3. [3] Abels H., On the notion of generalized solutions of two-phase flows for viscous incompressible fluids, RIMS Kôkyûroku BessatsuB1 (2007) 1-15. MR2312912
  4. [4] Ambrosio L., Fusco N., Pallara D., Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Clarendon Press, Oxford, 2000, xviii, p. 434. Zbl0957.49001MR1857292
  5. [5] Boyer F., Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal.20 (2) (1999) 175-212. Zbl0937.35123MR1700669
  6. [6] Chen X., Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom.44 (2) (1996) 262-311. Zbl0874.35045MR1425577
  7. [7] Denisova I.V., Solonnikov V.A., Solvability in Hölder spaces of a model initial–boundary value problem generated by a problem on the motion of two fluids, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)188 (1991) 5-44, Funktsii. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor.22 (1991) 5-44, 186. Zbl0756.35067MR1111467
  8. [8] Edwards R.E., Functional Analysis, Dover Publications, Inc., New York, 1995, 783 p. Zbl0182.16101MR1320261
  9. [9] Evans L.C., Gariepy R.F., Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992. Zbl0804.28001MR1158660
  10. [10] Gurtin M.E., Polignone D., Viñals J., Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci.6 (6) (1996) 815-831. Zbl0857.76008MR1404829
  11. [11] Hoffmann K.-H., Starovoitov V.N., Phase transitions of liquid–liquid type with convection, Adv. Math. Sci. Appl.8 (1) (1998) 185-198. Zbl0958.35152MR1623346
  12. [12] Hohenberg P., Halperin B., Theory of dynamic critical phenomena, Rev. Modern Phys.49 (1977) 435-479. 
  13. [13] Kim N., Consiglieri L., Rodrigues J.F., On non-Newtonian incompressible fluids with phase transitions, Math. Methods Appl. Sci.29 (13) (2006) 1523-1541. Zbl1101.76004MR2249576
  14. [14] Liu C., Shen J., A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D179 (3–4) (2003) 211-228. Zbl1092.76069MR1984386
  15. [15] Luckhaus S., Sturzenhecker T., Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations3 (2) (1995) 253-271. Zbl0821.35003MR1386964
  16. [16] Maekawa Y., On a free boundary problem for viscous incompressible flows, Interfaces Free Bound.9 (4) (2007) 549-589. Zbl1132.76303MR2358216
  17. [17] Modica L., The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal.98 (2) (1987) 123-142. Zbl0616.76004MR866718
  18. [18] Modica L., Mortola S., Un esempio di Γ - -convergenza, Boll. Unione Mat. Ital. B (5)14 (1) (1977) 285-299. Zbl0356.49008MR445362
  19. [19] Plotnikov P., Generalized solutions to a free boundary problem of motion of a non-Newtonian fluid, Siberian Math. J.34 (4) (1993) 704-716. Zbl0814.76007MR1248797
  20. [20] Röger M., Solutions for the Stefan problem with Gibbs–Thomson law by a local minimisation, Interfaces Free Bound.6 (1) (2004) 105-133. Zbl1050.35155MR2047075
  21. [21] Schätzle R., Hypersurfaces with mean curvature given by an ambient Sobolev function, J. Differential Geom.58 (3) (2001) 371-420. Zbl1055.49032MR1906780
  22. [22] Simon J., Compact sets in the space L p ( 0 , T ; B ) , Ann. Mat. Pura Appl. (4)146 (1987) 65-96. Zbl0629.46031MR916688
  23. [23] Simon L., Lectures on geometric measure theory, vol. 3, in: Proceedings of the Centre for Mathematical Analysis, Australian National University, Australian National University Centre for Mathematical Analysis, Canberra, 1983. Zbl0546.49019MR756417
  24. [24] Sohr H., The Navier–Stokes equations, in: Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001, an elementary functional analytic approach. Zbl0983.35004MR1928881

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.