A compactness lemma of aubin type and its application to degenerate parabolic equations

Abstract

Let Ω ⊂ Rⁿ be a regular domain and Φ(s) ∈ C loc (R) be a given function. If M⊂ L₂ (0, T ; W½ (Ω)) ∩ L ∞ (Ω × (0, T )) is bounded and the set {∂t Φ(v)|v ∈ M} is bounded in L₂ (0, T ; W-¹₂ (Ω)), then there is a sequence {vk} ∈ M such that vk ⇀ v ∈ L₂ (0,T ; W¹₂ (Ω)), and vk → v, Φ(vk) → Φ(v) a.e. in Ωτ = Ω × (0, T). This assertion is applied to prove solvability of the one-dimensional initial and boundary-value problem for a degenerate parabolic equation arising in the Buckley-Leverett model of two-phase filtration. We prove existence and uniqueness of a weak solution, establish the property of finite speed of propagation and construct a self-similar solution