Well-posedness of the Hele–Shaw–Cahn–Hilliard system

Wang, Xiaoming; Zhang, Zhifei

doi:10.1016/j.anihpc.2012.06.003

Wang, Xiaoming ; Zhang, Zhifei

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 367-384.

Résumé

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 $C ([0, T]; H^{s}) \cap L^{2} (0, T; H^{s + 2})$ 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.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2012.06.003

Mots clés : 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{\textendash}Shaw{\textendash}Cahn{\textendash}Hilliard} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {367--384},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.06.003},
     mrnumber = {3061427},
     zbl = {1291.35240},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.06.003/}
}

TY  - JOUR
AU  - Wang, Xiaoming
AU  - Zhang, Zhifei
TI  - Well-posedness of the Hele–Shaw–Cahn–Hilliard system
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 367
EP  - 384
VL  - 30
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.06.003/
DO  - 10.1016/j.anihpc.2012.06.003
LA  - en
ID  - AIHPC_2013__30_3_367_0
ER  -

%0 Journal Article
%A Wang, Xiaoming
%A Zhang, Zhifei
%T Well-posedness of the Hele–Shaw–Cahn–Hilliard system
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 367-384
%V 30
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.06.003/
%R 10.1016/j.anihpc.2012.06.003
%G en
%F 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, Tome 30 (2013) no. 3, pp. 367-384. doi : 10.1016/j.anihpc.2012.06.003. http://www.numdam.org/articles/10.1016/j.anihpc.2012.06.003/

Bibliographie
Cité par

[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 | Zbl

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

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

[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

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

[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 | EuDML | Numdam | MR | Zbl

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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

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

Cité par Sources :