Homogenization of Degenerate Porous Medium Type Equations in Ergodic Algebras

Abstract: We consider the homogenization problem for general porous medium type equations of the form $u_t=\D f(x,\frac{x}{\ve}, u)$. The pressure function $f(x,y,\cdot)$ may be of two different types. In the type~1 case, $f(x,y,\cdot)$ is a general strictly increasing function; this is a mildly degenerate case. In the type~2 case, $f(x,y,\cdot)$ has the form $h(x,y)F(u)+S(x,y)$, where $F(u)$ is just a nondecreasing function; this is a strongly degenerate case. We address the initial-boundary value problem for a general, bounded or unbounded, domain $\Om$, with null (or, more generally, steady) pressure condition on the boundary. The homogenization is carried out in the general context of ergodic algebras. As far as the authors know, homogenization of such degenerate quasilinear parabolic equations is addressed here for the first time. We also review the existence and stability theory for such equations and establish new results needed for the homogenization analysis. Further, we include some new results on algebras with mean value, specially a new criterion establishing the null measure of level sets of elements of the algebra, which is useful in connection with the homogenization of porous medium type equations in the type~2 case.