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Homogenization of degenerate porous medium type equations in ergodic algebras. (English) Zbl 1284.35049

Summary: We consider the homogenization problem for general porous medium type equations of the form \(u_t=\Delta f(x,\frac{x}{\varepsilon}, u)\). 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 \({\varOmega}\), 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.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35L65 Hyperbolic conservation laws
35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
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