A Compactness Theorem for Homogenization of Parabolic Partial Differential Equations

It is well known that the modeling of physical processes in strongly inhomogeneous media leads to the study of differential equations with rapidly varying coefficients. Regarding coefficients as periodic functions, many attempts for getting approximate solutions are accomplished, and some of successful ones are G-convergence by Spagnolo, H-convergence by Tartar and Γ-convergence by De Giorgi. Another common way is to use formal asymptotic expansion we first guess by a formal expansion what the limit should be and then justify it by energy method. Two-scale convergence defined by G. Allaire ([1]) is an efficient way of combining these two procedures two-scale convergence guarantees the least amount of convergent degree, which is stronger than weak convergence and weaker than norm convergence. But it is restricted to the elliptic cases. For studying parabolic differential equations, the convergent nature is not clear and even two-scale convergence can not be applied directly. Even though stationary problems corresponding to the given parabolic problems may be considered for homogenization process, the convergent relationship between stationary problems and non-stationary problems is not justified by the two-scale convergence defined by G. Allaire. In this note, we present a new