Electron. J. Diff. Equ., Vol. 2014 (2014), No. 227, pp. 1-13.

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

Anvarbek Meirmanov, Sergey Shmarev

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 ${ m M}\subset L_2(0,T;W^1_2(\Omega)) \cap L_{\infty}(\Omega\times (0,T))$ is bounded and the set $\{\partial_t\Phi(v)|\,v\in { m M}\}$ is bounded in $L_2(0,T;W^{-1}_2(\Omega))$, then there is a sequence $\{v_k\}\in { m M}$ 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.

Submitted September 25, 2014. Published October 27, 2014.
Math Subject Classifications: 35B27, 46E35, 76R99.
Key Words: Compactness lemma; two-phase filtration; nonlinear PDE; degenerate parabolic equations.

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Anvarbek Meirmanov
Department of mahtematics, Belgorod State University
ul.Pobedi 85, 308015 Belgorod, Russia
email: anvarbek@list.ru
Sergey Shmarev
Department of Mathematics, University of Oviedo
c/Calvo Sotelo s/n, 33007, Oviedo, Spain
email: shmarev@uniovi.es

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