We study the homogenization of a certain linear hyperbolic-parabolic problem exhibiting two rapid spatial scales {ε, ε2}. The homogenization is performed by means of evolution multiscale convergence, a generalization of the concept of two-scale convergence to include any number of scales in both space and time. In particular we apply a compactness result for gradients. The outcome of the homogenization procedure is that we obtain a homogenized problem of hyperbolic-parabolic type together with two elliptic local problems, one for each rapid scale.

We study the homogenization of a hyperbolic-parabolic PDE with oscillations in one fast spatial scale. Moreover, the first order time derivative has a degenerate coefficient passing to infinity when epsilon -> 0. We obtain a local problem which is of elliptic type, while the homogenized problem is also in some sense an elliptic problem but with the limit for epsilon(-1) partial derivative(t)mu(epsilon). tue as an undetermined extra source term in the right-hand side. The results are somewhat surprising and work remains to obtain a fully rigorous treatment. Hence the last section is devoted to a discussion of the reasonability of our conjecture including numerical experiments.

Reconstructing the homogenized coefficient, which is also called the G-limit, in elliptic equations involving heterogeneous media is a typical nonlinear ill-posed inverse problem. In this work, we develop a numerical technique to determine G-limit that does not rely on any periodicity assumption. The approach is a technique that separates the computation of the deviation of the G-limit from the weak L-2-limit of the sequence of coefficients from the latter. Moreover, to tackle the ill-posedness, based on the classical Tikhonov regularization scheme we develop several strategies to regularize the introduced method. Various numerical tests for both standard and non-standard homogenization problems are given to show the efficiency and feasibility of the proposed method.

The main contribution of this paper is the homogenization of the linear parabolic equation ∂tuε(x,t)-∇·(a(x/εq1,...,x/εqn,t/εr1,...,t/εrm)∇uε(x,t))=f(x,t) exhibiting an arbitrary finite number of both spatial and temporal scales. We briefly recall some fundamentals of multiscale convergence and provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtain n local problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales with q1=1, q2=2, and 0

We consider the homogenization of the linear parabolic problem which exhibits a mismatch between the spatial scales in the sense that the coefficient of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

Noé Bárcenas Torres Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy . . . . . . . . . . . . . . . . . . 387 … Junfei Dai, Wei Luo, Min Zhang, Xianfeng Gu and Shing-Tung Yau Visualization of 2-Dimensional Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 … Xingyu Dai, Fang Li and Kefeng Liu Boltje-Maisch Resolutions of Specht Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 … L. Flodén, A. Holmbom, M. Olsson Lindberg and J. Persson Two-Scale Convergence. Some Remarks and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461  … Philippe Gille et Anne Quéguiner-Mathieu Exemples de Groupes Semi-Simples Simplement Connexes Anisotropes Contenant Un Sous-Groupe Unipotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 … Xiao-Liu Wang and Wei-Feng Wo On the Asymptotic Stability of Stationary Lines in the Curve Shortening Problem . . 493

We first study the fundamental ideas behind two-scale convergence to enhance an intuitive understanding of this notion. The classical definitions and ideas are motivated with geometrical arguments illustrated by illuminating figures. Then a version of this concept, very weak two-scale convergence, is discussed both independently and briefly in the context of homogenization. The main features of this variant are that it works also for certain sequences of functions which are not bounded in L-2 (Omega) and at the same time is suited to detect rapid oscillations in some sequences which are strongly convergent in L-2 (Omega). In particular, we show how very weak two-scale convergence explains in a more transparent way how the oscillations of the governing coefficient of the PDE to be homogenized causes the deviation of the G-limit from the weak L-2 (Omega)(NxN)-limit for the sequence of coefficients. Finally, we investigate very weak multiscale convergence and prove a compactness result for separated scales which extends a previous result which required well-separated scales.

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

We apply a new version of multiscale convergence named very weak multiscale convergence to find possible frequencies of oscillation in an unknown coecient of a partial dierential equation from its solution. We also use this notion to study homogenization of a certain linear parabolic problem with multiple spatial and temporal scales.

We briefly recall the concept of multiscale convergence, which is a generalization of two-scale convergence. Then we investigate a related concept, called very weak multiscale convergence, and prove a compactness result with respect to this type of convergence. Finally we illustrate how this result can be used to study homogenization problems with several scales of oscillations.

Reiterated homogenization is studied for divergence structure parabolic problems of the form \({\frac{\partial u_{\varepsilon}}{\partial t}} - \mathrm{div}\left(a\left({\frac{x}{\varepsilon}},{\frac{x}{\varepsilon^2}} ,t, D u_{\varepsilon}\right)\right)=f\) . It is shown that under standard assumptions on the function a(y 1,y 2,t,ξ) the sequence \(\{u_{\varepsilon}\}\) of solutions converges weakly in \( L^p(0,T;W^{1,p}_0(\Omega))\) to the solution u of the homogenized problem \({\frac{\partial u}{\partial t}} - \mathrm{div}\left( b \left( t,D u \right)\right) = f\) .

On the convergence and determination of limits for some sequences of differential operators : Presented at The Ninth International Conference on Integral Methods in Science and Engineering, Niagara Falls, Ontario, Canada, July 23-27 2006

A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L 2(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.

We introduce a version of two-scale convergence that deals with certain non-periodic cases still preserv- ing some of the key properties of two-scale convergence. Examples different from those in traditional periodic two- scale convergence are demonstrated and the relationship with other generalizations of two-scale convergence is discussed.