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, Volume 26 (2009) no. 6, p. 2403-2424
@article{AIHPC_2009__26_6_2403_0,
     author = {Abels, Helmut and R\"oGer, Matthias},
     title = {Existence of Weak Solutions for a Non-Classical Sharp Interface Model for a Two-Phase Flow of Viscous, Incompressible Fluids},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {26},
     number = {6},
     year = {2009},
     pages = {2403-2424},
     doi = {10.1016/j.anihpc.2009.06.002},
     zbl = {pre05649879},
     mrnumber = {2569901},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2009__26_6_2403_0}
}
Abels, Helmut; 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, Volume 26 (2009) no. 6, pp. 2403-2424. doi : 10.1016/j.anihpc.2009.06.002. http://www.numdam.org/item/AIHPC_2009__26_6_2403_0/

[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. | MR 2563636 | Zbl pre05640833

[2] Abels H., On Generalized Solutions of Two-Phase Flows for Viscous Incompressible Fluids, Interfaces Free Bound. 9 (2007) 31-65. | MR 2317298 | Zbl 1124.35060

[3] Abels H., On the Notion of Generalized Solutions of Two-Phase Flows for Viscous Incompressible Fluids, RIMS Kôkyûroku Bessatsu B1 (2007) 1-15. | MR 2312912 | Zbl 1119.35042

[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. | MR 1857292 | Zbl 0957.49001

[5] Boyer F., Mathematical Study of Multi-Phase Flow Under Shear Through Order Parameter Formulation, Asymptot. Anal. 20 (2) (1999) 175-212. | MR 1700669 | Zbl 0937.35123

[6] Chen X., Global Asymptotic Limit of Solutions of the Cahn-Hilliard Equation, J. Differential Geom. 44 (2) (1996) 262-311. | MR 1425577 | Zbl 0874.35045

[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. | MR 1111467 | Zbl 0756.35067

[8] Edwards R. E., Functional Analysis, Dover Publications, Inc., New York, 1995, 783 p. | MR 1320261 | Zbl 0182.16101

[9] Evans L. C., Gariepy R. F., Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992. | MR 1158660 | Zbl 0804.28001

[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. | MR 1404829 | Zbl 0857.76008

[11] Hoffmann K.-H., Starovoitov V. N., Phase Transitions of Liquid-Liquid Type With Convection, Adv. Math. Sci. Appl. 8 (1) (1998) 185-198. | MR 1623346 | Zbl 0958.35152

[12] Hohenberg P., Halperin B., Theory of Dynamic Critical Phenomena, Rev. Modern Phys. 49 (1977) 435-479.

[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. | MR 2249576 | Zbl 1101.76004

[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. D 179 (3-4) (2003) 211-228. | MR 1984386 | Zbl 1092.76069

[15] Luckhaus S., Sturzenhecker T., Implicit Time Discretization for the Mean Curvature Flow Equation, Calc. Var. Partial Differential Equations 3 (2) (1995) 253-271. | MR 1386964 | Zbl 0821.35003

[16] Maekawa Y., On a Free Boundary Problem for Viscous Incompressible Flows, Interfaces Free Bound. 9 (4) (2007) 549-589. | MR 2358216 | Zbl 1132.76303

[17] Modica L., The Gradient Theory of Phase Transitions and the Minimal Interface Criterion, Arch. Ration. Mech. Anal. 98 (2) (1987) 123-142. | MR 866718 | Zbl 0616.76004

[18] Modica L., Mortola S., Un Esempio Di Γ - -Convergenza, Boll. Unione Mat. Ital. B (5) 14 (1) (1977) 285-299. | MR 445362 | Zbl 0356.49008

[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. | MR 1248797 | Zbl 0814.76007

[20] Röger M., Solutions for the Stefan Problem With Gibbs-Thomson Law by a Local Minimisation, Interfaces Free Bound. 6 (1) (2004) 105-133. | MR 2047075 | Zbl 1050.35155

[21] Schätzle R., Hypersurfaces With Mean Curvature Given by an Ambient Sobolev Function, J. Differential Geom. 58 (3) (2001) 371-420. | MR 1906780 | Zbl 1055.49032

[22] Simon J., Compact Sets in the Space L p (0,T;B), Ann. Mat. Pura Appl. (4) 146 (1987) 65-96. | MR 916688 | Zbl 0629.46031

[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. | MR 756417 | Zbl 0546.49019

[24] Sohr H., The Navier-Stokes Equations, in: Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001, an elementary functional analytic approach. | MR 1928881 | Zbl 0983.35004