Existence of Weak Solutions for a Non-Classical Sharp Interface Model for a Two-Phase Flow of Viscous, Incompressible Fluids

Abels, Helmut; RöGer, Matthias

doi:10.1016/j.anihpc.2009.06.002

Abels, Helmut ; RöGer, Matthias

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 6, pp. 2403-2424

MR | 2 citations dans Numdam

DOI : 10.1016/j.anihpc.2009.06.002

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     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},
     pages = {2403--2424},
     year = {2009},
     publisher = {Elsevier},
     volume = {26},
     number = {6},
     doi = {10.1016/j.anihpc.2009.06.002},
     mrnumber = {2569901},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2009.06.002/}
}

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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, Tome 26 (2009) no. 6, pp. 2403-2424. doi: 10.1016/j.anihpc.2009.06.002

Bibliographie
Cité par

[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

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

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

[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. | Zbl | MR

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

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

[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. | Zbl | MR

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

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

[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. | Zbl | MR

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

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

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

[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. | Zbl | MR

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

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