Infiltration Equation with Degeneracy on the Boundary

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Infiltration Equation with Degeneracy on the Boundary Huashui Zhan1

Received: 25 January 2017 / Accepted: 12 August 2017 © Springer Science+Business Media B.V. 2017

Abstract This paper is mainly about the infiltration equation   ut = div a(x)|u|α |∇u|p−2 ∇u , (x, t) ∈  × (0, T ), where p > α > 0, a(x) ∈ C 1 (), a(x) ≥ 0 with a(x)|x∈∂ = 0. If there is a constant β  1,−β such that  a (x)dx ≤ c, p > 1+ β1 , then the weak solution is smooth enough to define the trace on the boundary,the stability of the weak solutions can be proved as usual. Meanwhile, 1 ,  a −β (x)dxdt = ∞, then the weak solution lacks the regularity to if for any β > p−1 define the trace on the boundary. The main innovation of this paper is to introduce a new kind of the weak solutions. By these new definitions of the weak solutions, one can study the stability of the weak solutions without any boundary value condition. Keywords Infiltration equation · Weak solution · Boundary degeneracy · Stability Mathematics Subject Classification 35K65 · 35K92 · 35K85 · 35R35

1 Introduction In the study of water infiltration through porous media, Darcy’s linear relation V = −K(θ )∇φ,

(1.1)

satisfactorily describes the flow conduction provided that the velocities are small. Here V represents the seepage velocity of water, θ is the volumetric moisture content, K(θ ) is the The paper is supported by Natural Science Foundation of China (no: 11371297), Natural Science Foundation of Fujian province (no: 2015J01592), supported by Science Foundation of Xiamen University of Technology, China.

B H. Zhan

[email protected]; [email protected]

1

School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China

H. Zhan

hydraulic conductivity and φ is the total potential, which can be expressed as the sum of a hydrostatic potential ψ(θ ) and a gravitational potential z φ = ψ(θ ) + z. However, (1.1) fails to describe the flow for large velocities. To get a more accurate description of the flow in this case, several nonlinear versions of (1.1) have been proposed. One of these versions is V α = −K(θ )∇φ,

(1.2)

where α is a positive constant. If it is assumed that infiltration takes place in a horizontal column of the medium, then the continuity equation has the form ∂V ∂θ + = 0. ∂t ∂x Then we have  ∂  ∂θ = D(θ )p |θx |p−1 θx , ∂t ∂x

(1.3)

with p1 = α and D(θ ) = K(θ )ψ  (θ ). Considering the flows in fractured media, let ε be the size ratio of the matrix blocks to the whole medium and let the width of the fracture planes and the porous block diameter be in the same order. If the permeability ratio of matrix blocks to fracture planes is of order ε pε , where pε is a positive oscillating constant, then the nonlinear Darcy law combined with the continuity equation leads to the following equation pε −2 ε    ∇u = 0, ωε uεt − div k ε (x)∇uε 

(1.4)

where uε is the density of the fluid (which is generally denoted as ρ in other references), ωε , k ε are the porosity and the permeability of the medium. One can generalize Eqs. (1.3)

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