A vanishing dynamic capillarity limit equation with discontinuous flux
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
A vanishing dynamic capillarity limit equation with discontinuous flux M. Graf, M. Kunzinger, D. Mitrovic
and D. Vujadinovic
Abstract. We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation
∂t uε,δ + divfε,δ (x, uε,δ ) = εΔuε,δ + δ(ε)∂t Δuε,δ , x ∈ M, t ≥ 0 u|t=0 = u0 (x).
Here, fε,δ and u0 are smooth functions while ε and δ = δ(ε) are fixed constants. Assuming fε,δ → f ∈ Lp (Rd × R; Rd ) for some 1 < p < ∞, strongly as ε → 0, we prove that, under an appropriate relationship between ε and δ(ε) depending on the regularity of the flux f, the sequence of solutions (uε,δ ) strongly converges in L1loc (R+ × Rd ) toward a solution to the conservation law ∂t u + divf(x, u) = 0. The main tools employed in the proof are the Leray–Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept. Mathematics Subject Classification. 35K65, 42B37, 76S99. Keywords. Pseudo-parabolic equations, Vanishing dynamic capillarity, Discontinuous flux, Conservation laws.
1. Introduction and notation Flow in a two-phase porous medium is governed by the Darcy law [7] q = −K(S) (∇p + ρged ) ,
(1.1)
where ed = (0, . . . , 0, 1), is the direction of gravity. The quantity S is the saturation, p is the pressure, and (the vector) q is the flow velocity of the wetting phase (usually water, while the non-wetting one is oil or a gas). The Darcy law represents conservation of momentum and, in order to close the system, we also need the conservation of mass ∂t S + divq = 0.
(1.2)
In the two-dimensional situation, we have three equations, given by (1.1) and (1.2), while we have four unknowns (two velocity components, saturation, and pressure). Therefore, usually one assumes a constitutive relation between the pressure p, the capillary pressure Pc (equal to differences of pressures between wetting and non-wetting phases), and the saturation S. If it is assumed that the capillary pressure is 0123456789().: V,-vol
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(almost) constant, one can derive the Buckley–Leverett equation (a scalar conservation law derived in [8]): ∂t S + ∂x f (S) = 0,
(1.3)
2
S for f (S) = S 2 +A(1−S) 2 and an appropriate constant A. If we assume that Pc is “static” (independent of the t-derivative of S) in the sense that pc = pc (S), then we arrive at a parabolic perturbation of the Buckley–Leverett equation [7]. However, both of the models (the standard Buckley–Leverett or the one perturbed by a parabolic term) appear to give results inconsistent with certain fairly simple experiments [14]. Namely, if we take a thin tube filled with dry sand and dip it in water at a constant rate at one side of the tube and then measure the concentration of t
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