A Strange Term in the Homogenization of Parabolic Equations with Two Spatial and Two Temporal Scales

Abstract

We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficient a x/ε, t/ε 2 in the elliptic part and spatial oscillations in the coefficient ρ x/ε that is multiplied with the time derivative ∂ t u ε . We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation in ρ x/ε and the temporal oscillation in a x/ε, t/ε 2 and disappears if either of these oscillations is removed.

Self-excited Oscillations and Fuel Control of a Combustion Process in a Rijke Tube

“…3b, respectively. By applying numerical integration of solvability conditions (16), one obtains the slowly varying courses of amplitudes A 1 (τ 3 ), A 2 (τ 3 ), which are shown in Figs. 3a and 3b with dashed lines.…”

“…(15) and assuming K 1 = K 2 = 0 in Eq. (16) to consider the uncontrolled case, the solvability conditions of the first order are simplified in the form:…”

“…The spatial scales are not present in the model, because it is assumed that all heat is released into the fluid at a point [27]. Inclusion of spatial and temporal scales is considered in various homogenization problems, which encompasses the asymptotic analysis of solutions of PDEs of periodic structures [4], [16], [25]. Asymptotic expansion is used to derive the homogenized equation, however the method is formal and the second step is needed to prove the convergence as the ratio of the size of the period to the size of a sample of the medium goes to zero [1].…”

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

“…Following the procedure in Section 23.9 in [18], we obtain that { } is bounded in 2 (0, ; 1 0 (Ω)), see also [22]. Hence, (14) holds up to a subsequence. We proceed by studying the weak form of (12); that is,…”

“…Concerning cases where, as in (1) above, we do not have ≡ 1, Nandakumaran and Rajesh [12] studied a nonlinear parabolic problem with the same frequency of oscillation in time and space, respectively, in the elliptic part of the equation and an operator oscillating in space with the same frequency appearing in the temporal differentiation term. Recently, a number of papers have addressed various kinds of related problems where the temporal scale is not assumed to be identical with the spatial scale, see for example, [13,14]. Up to the authors' knowledge, this is the first study of this type of problems where the oscillations of the coefficient of the term including the time derivative do not match the spatial oscillations of the elliptic part.…”