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

Abstract

Let $\Omega\subset \mathbb{R}^{n}$ be a regular domain and $\Phi(s)\in C_{\rm loc}(\mathbb{R})$ be a given function. If $\mathfrak{M}\subset L_2(0,T;W^1_2(\Omega)) \cap L_{\infty}(\Omega\times (0,T))$ is bounded and the set $\{\partial_t\Phi(v)|\,v\in \mathfrak{M}\}$ is bounded in $L_2(0,T;W^{-1}_2(\Omega))$, then there is a sequence $\{v_k\}\in \mathfrak{M}$ such that $v_k\rightharpoonup v \in L^2(0,T;W^1_2(\Omega))$, and $v_k\to v$, $\Phi(v_k)\to \Phi(v)$ a.e. in $\Omega_T=\Omega\times (0,T)$. 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.

Keywords

• Compactness lemma
• two-phase filtration
• nonlinear PDE
• degenerate parabolic equations