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Global classical solution of Muskat free boundary problem. (English) Zbl 1038.35083
Summary: The Muskat problem which describes a two-phase flow of two fluids, for example, oil and water, in porous media is discussed. The problem involves in seeking two time-dependent harmonic functions \(u_1(x,y,t)\) and \(u_2(x,y,t)\) in oil and water regions, respectively, and the interface between oil and water, i.e., the free boundary \(\varGamma: y=\rho (x,t)\), such that on the free boundary \[ u_1=u_2,\;\;\;V_n=-k_1 \frac{\partial u_1}{\partial n}=-k_2 \frac {\partial u_2}{\partial n} \] where \(n\) the unit normal vector on the free boundary toward oil region, \(V_n\) is the normal velocity of the free boundary \(\varGamma\), \(k_1\) and \(k_2\) are positive constants satisfying \(k_1 > k_2\). We prove the existence of classical solution globally in time under some reasonable assumptions. The argument developed in this paper can be used in any multidimensional case.

35Q35 PDEs in connection with fluid mechanics
76S05 Flows in porous media; filtration; seepage
35R35 Free boundary problems for PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
Full Text: DOI
[1] Muskat, M., Two fluid systems in porous media. the encroachment of water into an oil sand, Physics, 5, 250-264, (1934) · JFM 60.1388.01
[2] Jiang, L.; Chen, Y., Weak formulation of multidimemsional Muskat problem, (), 509-513
[3] Schroll, H.J.; Tveito, A., Local existence and stability for a hyperbolic – elliptic system modeling two-phase reservoir flow, Electron. J. differential equations, 2000, 1-28, (2000) · Zbl 0938.35005
[4] Kametaka, Y.; Radkevich, E., Passage to the limit in a chain of the Muskat problem (I), Appl. anal., 71, 91-109, (2000) · Zbl 1021.35050
[5] Kametaka, Y.; Radkevich, E., Passage to the limit in a chain of the Muskat problem (II), Appl. anal., 71, 279-299, (2000) · Zbl 1042.35621
[6] Yi, F., Local classical solution of Muskat free boundary problem, J. partial differential equations, 9, 84-96, (1996) · Zbl 0847.35152
[7] Xu, L.; Zhang, J., Classical solution of Muskat problem with kinetic, Pure appl. math., 18, 332-337, (2002), in Chinese · Zbl 1128.35391
[8] Radkevich, E., On conditions for the existence of a classical solution of the modified Stefan problem (the gibbs – thomson law), Russian acad. sci. math., 75, 221-246, (1993) · Zbl 0772.35087
[9] Jiang, L.; Liang, J., The perturbation of the interface of the two-dimensional diffraction problem and an approximating Muskat problem, J. partial differential equations, 3, 85-96, (1990) · Zbl 0708.35095
[10] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, Progress in nonlinear differential equations and their applications, 16, (1995) · Zbl 0816.35001
[11] Simon, J., Compact sets in the space Lp(0,T;B), Ann. mat. pura appl., 146, 65-96, (1987) · Zbl 0629.46031
[12] J.M. Bony, Analyse microlocale des equations aux derivees partielles non lineaires, in: Lecture Notes in Math., Vol. 1495 · Zbl 0758.35004
[13] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer-Verlag · Zbl 0691.35001
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