A Linear Degenerate Elliptic Equation Arising from Two-Phase Mixtures

Abstract

We consider the linear degenerate elliptic system of two first order equations ${\mathbf{u}}=-a(\phi)(\nabla{p} - \mathbf{g})$ and $\nabla\cdot(b(\phi)\mathbf{u})+\phi{p}=\phi^{1/2}{f}$, where $a$ and $b$ satisfy $a(0)=b(0)=0$ and are otherwise positive, and the porosity $\phi\ge0$ may be zero on a set of positive measure. This model equation has a similar degeneracy to that arising in the equations describing the mechanical system modeling the dynamics of partially melted materials, e.g., in the earth's mantle and in polar ice sheets and glaciers. In the context of mixture theory, $\phi$ represents the phase variable separating the solid one-phase ($\phi=0$) and fluid-solid two-phase ($\phi>0$) regions. The equations should remain well-posed as $\phi$ vanishes so that the free boundary between the one- and two-phase regions need not be found explicitly. Two main problems arise. First, as $\phi$ vanishes, one equation is lost. Second, it is shown by stability or energy bounds for the solution that the pre...