A compactness lemma of Aubin type and its application to degenerate parabolic equations - Anvarbek Meirmanov; Sergey Shmarev (Electronic Journal of Differential Equations, 2014)
Let $Ω\subset \mathbb{R}^{n}$ be a regular domain and $Φ(s)\in C_{\rm loc}(\mathbb{R})$ be a given function. If $\mathfrak{M}\subset L_2(0,T;W^1_2(Ω)) \cap L_{∞}(Ω× (0,T))$ is bounded and the set $\{\partial_tΦ(v)|\,v\in \mathfrak{M}\}$ is bounded in $L_2(0,T;W^{-1}_2(Ω))$, then there is a sequence $\{v_k\}\in \mathfrak{M}$ such that $v_k\rightharpoonup v \in L^2(0,T;W^1_2(Ω))$, and $v_k→ v$, $Φ(v_k)→ Φ(v)$ a.e. in $Ω_T=Ω× (0,T)$. 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.