Qualitative Analysis of the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption
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The goal of the dissertation is to pursue qualitative analysis of the mathematical model of turbulent polytropic filtration of a gas in a porous media with reaction or absorption described by the second order nonlinear double degenerate parabolic equation ∂u ∂t − ∂ ∂x F [ ∂u m ∂x ] + Q(u) = 0, (1) where F(y) = |y| p−1 y, Q(u) = buβ , m, p, β > 0, b ∈ R. In the absence of the reaction term there is a finite speed of propagation with an expanding interface in the case of slow diffusion (mp > 1), and infinite speed of propagation in the case of fast diffusion (0 < mp < 1). In general, qualitative properties of the turbulent filtration is an outcome of the competition between the diffusion and reaction or absorption forces. In the slow diffusion case, the strong domination of the diffusion causes an expanding interface. In the fast diffusion case, the strong domination of the diffusion causes infinite speed of propagation and absence of interfaces, while weak domination of the diffusion causes an expanding interface. Domination of the absorption causes a shrinking interface. If diffusion and absorption are in balance then initial density profile dictates the direction of the interface movement. When the interface exists, explicit asymptotic formulas and estimates for the interface function and the local solution near the interfaces with accuracy up to precise constants are proved. Explicit asymptotic formulas for the local solution at infinity are proved in all cases where interface does not exist. The results of the dissertation can be applied to problems in the oil and gas industry to pursue the estimation and control of the time evolution of the size of oil and gas resources.