Homogenization of Parabolic Equations with Non-self-similar Scales

Abstract

This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in different scales. We obtain large-scale interior and boundary Lipschitz estimates as well as interior C 1,α and C 2,α estimates by utilizing higher-order correctors. We also investigate the problem of convergence rates for initial-boundary value problems.MSC2010: 35B27, 35K40.

Compactness and stable regularity in multiscale homogenization

“…Let us first consider a slightly simpler case A ε (x, t) = A(x, t, x/ε, t/δ 2 ) with arbitrary (ε, δ) ∈ (0, 1] 2 , and A(x, t, y, s) is periodic in (y, s) ∈ R d × R. We recall that the special cases A ε (x, t) = A(x/ε, t/ε k ) with k > 0, and A ε (x, t) = A(x/ε, t/δ 2 ) with arbitrary (ε, δ) ∈ (0, 1] 2 have been studied in [17] and [16], respectively. In particular, the Lipschitz estimate uniform in (ε, δ) was derived in [16] by the access decay method combined with a proper rescaling argument.…”

“…Let u λ ε be a weak solution of −div(A λ (x, x/ε 1 )∇u λ ε ) = 0 in B 2r , and set δ = max 1≤i≤d λ −1 i ε 1 ≤ r ≤ 1. By performing the same analysis as Section 3 in [16] (see [17] and [22] for the original ideas), one can prove that there exists a weak solution…”

“…A simple example of reperiodization may be seen in Subsection 1.4. This method can be adapted easily to parabolic equations with non-self-similar scales (for time-dependent coefficients) and gives simple proofs to the regularity estimates contained in [17,16]; see Theorem 4.5.…”

“…The second approach (also motivated by the recent work [17,16]) is based on the excess decay method developed recently in [5,26]. One of the key step is to find proper smooth functions to approximate the solution u ε of the original oscillating problem.…”

“…The key point is then to show that the convergence rate to the effective solutions is stable with respect to (ε i ) 1≤i≤d ; see Theorem 4.6. This requires us to obtain some uniform estimates related to certain degenerate cell problems, which can be handled by the idea of [17,16] or the idea of reperiodization mentioned above. 1.4.…”