Homogenization of some parabolic operators with several time scales

A Note on Parabolic Homogenization with a Mismatch between the Spatial Scales

“…We prove by means of very weak multiscale convergence [2] that the corrector 2 associated with the gradient for the second rapid spatial scale 2 actually vanishes. Already, in [3,4], it was observed that having more than one rapid temporal scale in parabolic problems does not generate a reiterated problem and in this paper we can see that nor does the addition of a spatial scale if it is contained in a coefficient that is multiplied with the time derivative of .…”

Homogenization of monotone parabolic problems with several temporal scales

“…where the second equality follows from the fact that u i is Y i -periodic. Hence, by (14) and the well-known characterisation of the W 1 -norm in terms of an L 2 -norm of the gradient (see, e.g., Proposition 3.52 in [8]),…”

“…Homogenisation for linear parabolic problems with several temporal scales using the multiscale convergence technique was first achieved by Flodén and Olsson in 2007 (see [14]). This was a further development of the work by Holmbom in 1996 and 1997 (see [16] and [17], respectively) where two-scale convergence was employed to homogenise linear parabolic problems with both a spatial and a temporal microscale.…”

Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales

“…Provided that this generalized sequence of equations has, for each fixed ε > 0, a solution u ε (this will be accomplished in Section 3), we have in hand a sequence (u ε ) ε>0 , and we are interested in the asymptotic behavior as 0 < ε → 0, of u ε . The motivation of the present study is twofold: (1) Few works deal with homogenization of degenerated non-monotone parabolic operators with coefficients depending on the macroscopic variables (x, t) (to the best of our knowledge there is no published work which put together these three aspects); (2) Homogenization with more than one time scale has been considered for the first time by Flodén and Olsson in a more recent work [6], and seems to be very attractive and instructive. To fix ideas, in [6] Flodén and Olsson study the periodic homogenization problem for linear monotone parabolic operators.…”

“…The motivation of the present study is twofold: (1) Few works deal with homogenization of degenerated non-monotone parabolic operators with coefficients depending on the macroscopic variables (x, t) (to the best of our knowledge there is no published work which put together these three aspects); (2) Homogenization with more than one time scale has been considered for the first time by Flodén and Olsson in a more recent work [6], and seems to be very attractive and instructive. To fix ideas, in [6] Flodén and Olsson study the periodic homogenization problem for linear monotone parabolic operators. The novelty of our work lies on the non-monotonicity (because of the term a 0 (x, t, x/ε, t/ε, t/ε k , u ε , Du ε )) and the degeneracy of the operator under consideration.…”