# zbMATH — the first resource for mathematics

Well-posedness of the Hele-Shaw-Cahn-Hilliard system. (English) Zbl 1291.35240
Summary: 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.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D27 Other free boundary flows; Hele-Shaw flows 76S05 Flows in porous media; filtration; seepage 42B25 Maximal functions, Littlewood-Paley theory 35B44 Blow-up in context of PDEs
##### Keywords:
Hele-Shaw-Cahn-Hilliard; well-posedness; blow-up criterion
Full Text:
##### References:
 [1] Abels, H., On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194, 463-506, (2009) · Zbl 1254.76158 [2] Ambrose, D. M., Well-posedness of two-phase Hele-Shaw flow without surface tension, European J. Appl. Math., 15, 597-607, (2004) · Zbl 1076.76027 [3] Ambrose, D. M., Well-posedness of two-phase Darcy flow in 3D, Quart. Appl. Math., 65, 189-203, (2007) · Zbl 1147.35073 [4] Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30, 139-165, (1998) · Zbl 1398.76051 [5] Bear, J., Dynamics of fluids in porous media, (1988), Dover · Zbl 1191.76002 [6] Bony, J.-M., Calcul symbolique et propagation des singularitiés pour LES équations aux dérivées partielles non linéaires, Ann. Sci. Ec. Norm. Super., 14, 209-246, (1981) · Zbl 0495.35024 [7] Boyer, F., Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20, 2, 175-212, (1999) · Zbl 0937.35123 [8] Caffarelli, L.; Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171, 1913-1930, (2010) · Zbl 1204.35063 [9] Chemin, J.-Y., Perfect incompressible fluids, (1998), Oxford University Press New York [10] Constantin, P.; Pugh, M., Global solutions for small data to the Hele-Shaw problem, Nonlinearity, 6, 393-415, (1993) · Zbl 0808.35104 [11] Cordoba, A.; Cordoba, D.; Gancedo, F., Interface evolution: the Hele-Shaw and Muskat problems, Ann. of Math., 173, 1, 477-542, (2011) · Zbl 1229.35204 [12] E, W.; Palffy-Muhoray, P., Phase separation in incompressible systems, Phys. Rev. E, 55, R3844-R3846, (1997) · Zbl 1110.82304 [13] Escher, J.; Simonett, G., Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28, 1028-1047, (1997) · Zbl 0888.35142 [14] Escher, J.; Simonett, G., A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143, 267-292, (1998) · Zbl 0896.35142 [15] X. Feng, S. Wise, Approximation of the HSCH system, 2010, in preparation. [16] Hohenberg, P. C.; Halperin, B. I., Theory of dynamic critical phenomena, Rev. Modern Phys., 49, 435-479, (1977) [17] Howison, S. D., A note on the two-phase Hele-Shaw problem, J. Fluid Mech., 409, 243-249, (2000) · Zbl 0962.76028 [18] Joseph, D. D.; Renardy, Y. Y., Fundamentals of two-fluid dynamics, parts I and II, (1993), Springer-Verlag New York · Zbl 0784.76003 [19] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 891-907, (1988) · Zbl 0671.35066 [20] Lee, H.-G.; Lowengrub, J. S.; Goodman, J., Modeling pinchoff and reconnection in a Hele-Shaw cell. I. the models and their calibration, Phys. Fluids, 14, 492-513, (2002) · Zbl 1184.76316 [21] Lin, F.; Liu, C., Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48, 501-537, (1995) · Zbl 0842.35084 [22] Lin, F.; Liu, C., Existence of solutions for the ericksen-Leslie system, Arch. Ration. Mech. Anal., 154, 135-156, (2000) · Zbl 0963.35158 [23] Xu, X.; Zhao, L.; Liu, C., Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations, SIAM J. Math. Anal., 41, 2246-2282, (2010) · Zbl 1203.35191 [24] Majda, A.; Bertozzi, A., Vorticity and incompressible flow, (2002), Cambridge University Press Cambridge, UK · Zbl 0983.76001 [25] Saffman, P. G.; Taylor, G. I., 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, 312-329, (1958) · Zbl 0086.41603 [26] Siegel, M.; Caflisch, R.; Howison, S., Global existence, singular solutions, and ill-posedness for the Muskat problem, Comm. Pure Appl. Math., 57, 1374-1411, (2004) · Zbl 1062.35089 [27] Temam, R., Navier-Stokes equations, (1977), North-Holland Amsterdam · Zbl 0335.35077 [28] Triebel, H., Theory of function spaces, Monogr. Math., (1983), Birkhäuser Verlag Basel, Boston · Zbl 0546.46028 [29] Wise, S., Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44, 38-68, (2010) · Zbl 1203.76153 [30] Wise, S.; Lowengrub, J.; Frieboes, H.; Cristini, V., Three-dimensional multispecies nonlinear tumor growth I model and numerical method, J. Theoret. Biol., 253, 524-543, (2008) · Zbl 1398.92135 [31] Workman, J. T., End-point estimates and multi-parameter paraproducts on higher dimensional tori · Zbl 1201.42006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.