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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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##### References:
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