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On the equation of turbulent filtration in one-dimensional porous media. (English) Zbl 0613.76102

The authors consider the one-dimensional, turbulent, polytropic flow of a gas in a porous medium described by the Leibenson model (*) \(u_ t=(D(u,u_*))_ x,\quad t>0,\quad x\in {\mathbb{R}},\quad u(0,x)=u_ 0(x)\in L^ 1({\mathbb{R}}),\quad u_ 0\geq 0\) a.e. where, in the diffusion coefficient \(D(u,u_ x)=m^ pu^{p(m-1)}| u_ x|^{p-1}\) the positive constants p and m satisfy \(mp>1\). They prove the existence and uniqueness of a strong solution of (*), together with some regularity properties which imply that the free boundaries are Lipschitz-continuous, non-decreasing curves.
Reviewer: D.Polisevski

MSC:

76S05 Flows in porous media; filtration; seepage
76F99 Turbulence
35K65 Degenerate parabolic equations
76R99 Diffusion and convection
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