Global existence of strong solutions to a groundwater flow problem
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Global existence of strong solutions to a groundwater flow problem Xiangsheng Xu Abstract. In this paper, we the initial boundary value problem for the system Δv = ux1 , ut − div study (a|q| + m)I + (b − a) q⊗q ∇u = −∇u · q, where q = (−vx2 , vx1 )T , q ⊗ q = qqT . This problem has been proposed |q| as a model for a fluid flowing through a porous medium under the influence of gravity and hydrodynamic dispersion. 2 For each T > 0, we obtain a so-called strong solution (v, u) in the function space L∞ (0, T ; W 1,∞ (Ω) ), where Ω is a bounded domain in R2 . The key ingredient in our approach is the decomposition A2 = tr(A)A − det(A)I for any 2 × 2 symmetric matrix A. By exploring this decomposition, we are able to derive an equation of parabolic type for the function j (a|q| + m)I + (b − a) q⊗q ∇u · ∇u , j ≥ 1. With the aid of this equation, we obtain a uniform bound for ∇u. |q| Mathematics Subject Classification. Primary 35B45, 35B65, 35M33, 35Q92. Keywords. Groundwater flow, Matrix decomposition, De Giorgi iteration method.
1. Introduction Let Ω be a bounded domain in the x = (x1 , x2 ) plane with boundary ∂Ω and T any positive number. We study the problem ut − div
Δv = ux1 in ΩT ≡ Ω × (0, T ), ∇u = −∇u · q in ΩT ,
q⊗q |q| q⊗q (a|q| + m)I + (b − a) ∇u · ν = 0 on ΣT ≡ ∂Ω × (0, T ), |q| v = 0 on ΣT , u(x, 0) = u0 (x) on Ω, (a|q| + m)I + (b − a)
where
q=
−vx2 vx1
(1.1) (1.2) (1.3) (1.4) (1.5)
,
(1.6)
q ⊗ q = qqT , I is the 2 × 2 identity matrix, a, b, m are positive numbers with b > a, and ν is the unit outward normal to the ∂Ω. This system arises in the description of the movement of a fluid of variable density u through a porous medium under the influence of gravity and hydrodynamic dispersion [5]. The first equation (1.1) is derived from Darcy’s law, while the second equation (1.2) describes the mass balance. See [5,15] for details. In a slightly different form, problem (1.1)–(1.5) was studied by Su [20] using classical partial differential 0123456789().: V,-vol
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equation (PDE) methods. In [5] the problem was formulated as abstract evolution equations in Banach spaces. To describe the results there, we set q⊗q (1.7) D ≡ (a|q| + m)I + (b − a) |q| If D can be taken as a constant multiple of the identity matrix I, the resulting problem has a classical solution, while in the general case, only local existence of weak solutions in W 1,p (Ω) for some p > 1 was obtained. The global existence was thereby left as an open problem. The objective of this paper is to solve the problem left open in [5]. Before we precisely state our result, we make some preliminary observations. By the definition of q, we always have divq = 0.
(1.8)
Thus, ∇u · q = div(uq). Moreover,
q ⊗ q |q| ≤ |q|.
It is natural for us to define
q⊗q = 0 whenever q = 0. |q| Thus, the coefficient matrix D is well defined and satisfies (a|q| + m)|ξ|2 ≤ Dξ · ξ = (a|q| + m)|ξ|2 +
b−a (q · ξ)2 ≤ (b|q| + m)|ξ|2 for each ξ ∈ R2 . |q|
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