Well-posedness of the Hele–Shaw–Cahn–Hilliard system
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, p. 367-384

We study the well-posedness of the Hele–Shaw–Cahn–Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in ${H}^{s}$, $s>\frac{d}{2}+1$, the existence and uniqueness of solution in $C\left(\left[0,T\right];{H}^{s}\right)\cap {L}^{2}\left(0,T;{H}^{s+2}\right)$ that is global in time in the two dimensional case ($d=2$) and local in time in the three dimensional case ($d=3$) are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood–Paley theory in order to establish certain key commutator estimates.

DOI : https://doi.org/10.1016/j.anihpc.2012.06.003
Keywords: Hele–Shaw–Cahn–Hilliard, Well-posedness, Blow-up criterion
@article{AIHPC_2013__30_3_367_0,
author = {Wang, Xiaoming and Zhang, Zhifei},
title = {Well-posedness of the Hele--Shaw--Cahn--Hilliard system},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {30},
number = {3},
year = {2013},
pages = {367-384},
doi = {10.1016/j.anihpc.2012.06.003},
zbl = {1291.35240},
mrnumber = {3061427},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2013__30_3_367_0}
}

Wang, Xiaoming; Zhang, Zhifei. Well-posedness of the Hele–Shaw–Cahn–Hilliard system. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, pp. 367-384. doi : 10.1016/j.anihpc.2012.06.003. http://www.numdam.org/item/AIHPC_2013__30_3_367_0/

[1] 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 2563636 | Zbl 1254.76158

[2] D.M. Ambrose, Well-posedness of two-phase Hele–Shaw flow without surface tension, European J. Appl. Math. 15 (2004), 597-607 | MR 2128613 | Zbl 1076.76027

[3] D.M. Ambrose, Well-posedness of two-phase Darcy flow in 3D, Quart. Appl. Math. 65 (2007), 189-203 | MR 2313156 | Zbl 1147.35073

[4] D.M. Anderson, G.B. Mcfadden, A.A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech. 30 (1998), 139-165 | MR 1609626

[5] J. Bear, Dynamics of Fluids in Porous Media, Dover (1988) | Zbl 1191.76002

[6] J.-M. Bony, Calcul symbolique et propagation des singularitiés pour les équations aux dérivées partielles non linéaires, Ann. Sci. Ec. Norm. Super. 14 (1981), 209-246 | Numdam | MR 631751 | Zbl 0495.35024

[7] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal. 20 no. 2 (1999), 175-212 | MR 1700669 | Zbl 0937.35123

[8] L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. 171 (2010), 1913-1930 | MR 2680400 | Zbl 1204.35063

[9] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford University Press, New York (1998) | MR 1688875

[10] P. Constantin, M. Pugh, Global solutions for small data to the Hele–Shaw problem, Nonlinearity 6 (1993), 393-415 | MR 1223740 | Zbl 0808.35104

[11] A. Cordoba, D. Cordoba, F. Gancedo, Interface evolution: the Hele–Shaw and Muskat problems, Ann. of Math. 173 no. 1 (2011), 477-542 | MR 2753607 | Zbl 1229.35204

[12] W. E, P. Palffy-Muhoray, Phase separation in incompressible systems, Phys. Rev. E 55 (1997), R3844-R3846 | MR 1449369 | Zbl 1110.82304

[13] J. Escher, G. Simonett, Classical solutions of multidimensional Hele–Shaw models, SIAM J. Math. Anal. 28 (1997), 1028-1047 | MR 1466667 | Zbl 0888.35142

[14] J. Escher, G. Simonett, A center manifold analysis for the Mullins–Sekerka model, J. Differential Equations 143 (1998), 267-292 | MR 1607952 | Zbl 0896.35142

[15] X. Feng, S. Wise, Approximation of the HSCH system, 2010, in preparation.

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

[17] S.D. Howison, A note on the two-phase Hele–Shaw problem, J. Fluid Mech. 409 (2000), 243-249 | MR 1756390 | Zbl 0962.76028

[18] D.D. Joseph, Y.Y. Renardy, Fundamentals of Two-Fluid Dynamics, Parts I and II, Springer-Verlag, New York (1993) | MR 1200238

[19] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907 | MR 951744 | Zbl 0671.35066

[20] H.-G. Lee, J.S. Lowengrub, J. Goodman, Modeling pinchoff and reconnection in a Hele–Shaw cell. I. The models and their calibration, Phys. Fluids 14 (2002), 492-513 | MR 1878351 | Zbl 1184.76316

[21] F. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math. 48 (1995), 501-537 | MR 1329830 | Zbl 0842.35084

[22] F. Lin, C. Liu, Existence of solutions for the Ericksen–Leslie system, Arch. Ration. Mech. Anal. 154 (2000), 135-156 | MR 1784963 | Zbl 0963.35158

[23] X. Xu, L. Zhao, C. Liu, Axisymmetric solutions to coupled Navier–Stokes/Allen–Cahn equations, SIAM J. Math. Anal. 41 (2010), 2246-2282 | MR 2579713 | Zbl 1203.35191

[24] A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, UK (2002) | MR 1867882 | Zbl 0983.76001

[25] P.G. Saffman, G.I. Taylor, The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous fluid, Proc. R. Soc. Lond. Ser. A 245 (1958), 312-329 | MR 97227 | Zbl 0086.41603

[26] M. Siegel, R. Caflisch, S. Howison, Global existence, singular solutions, and ill-posedness for the Muskat problem, Comm. Pure Appl. Math. 57 (2004), 1374-1411 | MR 2070208 | Zbl 1062.35089

[27] R. Temam, Navier–Stokes Equations, North-Holland, Amsterdam (1977) | MR 603444 | Zbl 0335.35077

[28] H. Triebel, Theory of Function Spaces, Monogr. Math., Birkhäuser Verlag, Basel, Boston (1983) | MR 781540

[29] S. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn–Hilliard–Hele–Shaw system of equations, J. Sci. Comput. 44 (2010), 38-68 | MR 2647498 | Zbl 1203.76153

[30] S. Wise, J. Lowengrub, H. Frieboes, V. Cristini, Three-dimensional multispecies nonlinear tumor growth I model and numerical method, J. Theoret. Biol. 253 (2008), 524-543 | MR 2964567

[31] J.T. Workman, End-point estimates and multi-parameter paraproducts on higher dimensional tori, arXiv:0806.0197v1 | MR 2677008 | Zbl 1201.42006