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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.

MSC:
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
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