On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 6, pp. 1175-1190.

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain by allowing for a degenerate mobility. The model has been developed by Abels, Garcke and Grün for fluids with different densities and leads to a solenoidal velocity field. It is given by a non-homogeneous Navier–Stokes system with a modified convective term coupled to a Cahn–Hilliard system, such that an energy estimate is fulfilled which follows from the fact that the model is thermodynamically consistent.

DOI : 10.1016/j.anihpc.2013.01.002
Classification : 76T99, 35Q30, 35Q35, 76D03, 76D05, 76D27, 76D45
Mots clés : Two-phase flow, Navier–Stokes equations, Diffuse interface model, Mixtures of viscous fluids, Cahn–Hilliard equation, Degenerate mobility
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     title = {On an incompressible {Navier{\textendash}Stokes/Cahn{\textendash}Hilliard} system with degenerate mobility},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1175--1190},
     publisher = {Elsevier},
     volume = {30},
     number = {6},
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Abels, Helmut; Depner, Daniel; Garcke, Harald. On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 6, pp. 1175-1190. doi : 10.1016/j.anihpc.2013.01.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.01.002/

[1] H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Comm. Math. Phys. 289 (2009), 45-73 | MR | Zbl

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

[3] H. Abels, Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow, SIAM J. Math. Anal. 44 no. 1 (2012), 316-340 | MR | Zbl

[4] H. Abels, D. Depner, H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech. (2012), http://dx.doi.org/10.1007/s00021-012-0118-x, in press, arXiv:1111.2493.

[5] H. Abels, H. Garcke, G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci. 22 no. 3 (2012) | MR | Zbl

[6] G. Aki, W. Dreyer, J. Giesselmann, C. Kraus, A quasi-incompressible diffuse interface model with phase transition, WIAS preprint No. 1726, Berlin, 2012. | MR

[7] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal. 20 no. 2 (1999), 175-212 | MR | Zbl

[8] F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Comput. & Fluids 31 (2002), 41-68 | Zbl

[9] J.W. Cahn, J.E. Taylor, Surface motion by surface diffusion, Acta Metall. 42 (1994), 1045-1063

[10] L.-Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res. 32 (2002), 113-140

[11] J. Diestel, J.J. Uhl, Vector Measures, Amer. Math. Soc., Providence, RI (1977) | MR | Zbl

[12] H. Ding, P.D.M. Spelt, C. Shu, Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys. 22 (2007), 2078-2095 | Zbl

[13] C.M. Elliott, H. Garcke, On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal. 27 no. 2 (1996), 404-423 | MR | Zbl

[14] G. Grün, Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening, Z. Anal. Anwend. 14 (1995), 541-573 | MR | Zbl

[15] 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 no. 6 (1996), 815-831 | MR | Zbl

[16] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer (2001) | MR | Zbl

[17] J.E. Hilliard, Spinodal decomposition, Phase Transformations, American Society for Metals, Cleveland (1970), 497-560

[18] P.C. Hohenberg, B.I. Halperin, Theory of dynamic critical phenomena, Rev. Modern Phys. 49 (1977), 435-479

[19] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non linéaires, Dunod, Paris (1969) | MR | Zbl

[20] J. Lowengrub, L. Truskinovsky, Quasi-incompressible Cahn–Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), 2617-2654 | MR | Zbl

[21] I. Müller, Thermodynamics, Pitman Advanced Publishing Program, XVII, Boston–London–Melbourne, 1985.

[22] T. Roubíček, A generalization of the Lions–Temam compact embedding theorem, Časopis Pěst. Mat. 115 no. 4 (1990), 338-342 | EuDML | MR | Zbl

[23] J. Simon, Compact sets in the space L p (0,T;B), Ann. Mat. Pura Appl. (4) 146 (1987), 65-96 | MR | Zbl

[24] H. Sohr, The Navier–Stokes Equations, Birkhäuser Adv. Texts. Basler Lehrbücher, Birkhäuser Verlag, Basel (2001) | MR

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