On the well-posedness of generalized Darcy–Forchheimer equation

Abstract

Under the necessary compatibility condition and some mild regularity assumptions on the interior and the boundary data, we prove the existence, uniqueness, and

Buckley–Leverett Theory for a Forchheimer–Darcy Multiphase Flow Model with Phase Coupling

“…The proof of this proposition is similar to the proof of Proposition 3.1 in [24]. Inspired by Proposition 1, we write U = U g,w,0 + U g,w where U g,w,0 ∈ R g .…”

“…Proposition 2 (see Proposition 3.2 in Audu et al [24]). Problem (31) is equivalent to problem (23) and (24).…”

“…Lemma 2 (see Lemma 4.1 in Audu et al [24]). The operator A : X −→ Y satisfies the following inequalities:…”

“…Lemma 3 (see Lemma 4.2 in Audu et al [24]). The function U → A(U + U g,w ) defined in (32) is strongly monotone, that is, for all U, V ∈ X, we have…”

“…We refer to Proposition 4.1 in [24] for a proof. Indeed, summing-up and combining all the previous results, using the inf-sup condition in Lemma 1, and taking into consideration the linearity and continuity of the operator G, one establishes the existence of (U, P) solution of the system (9).…”