×

zbMATH — the first resource for mathematics

Oscillations and energy densities in the wave equation. (English) Zbl 0803.35010
From the introduction: We consider the following wave equation with Dirichlet boundary conditions: \[ \rho^ \varepsilon {\partial^ 2 u^ \varepsilon \over \partial t^ 2} - \text{div} (A^ \varepsilon \text{grad} u^ \varepsilon) = f \quad \text{in} \quad \Omega \times (0,T), \]
\[ u^ \varepsilon = 0 \quad \text{on} \quad \partial \Omega \times (0,T),\quad u^ \varepsilon (0) = a^ \varepsilon \quad \text{in} \quad \Omega,\quad {\partial u^ \varepsilon \over \partial t} (0) = b^ \varepsilon \quad \text{in} \quad \Omega, \] where \(u^ \varepsilon\) is the unknown field while the other quantities are given. Under “minimal” assumptions on the various data, it is possible to partition the solution \(u^ \varepsilon\) into \(u^ \varepsilon = \widetilde u^ \varepsilon + v^ \varepsilon\), where \(\widetilde u^ \varepsilon\) can be explicitly described from the only knowledge of the weak limit field \(u\) of \(u^ \varepsilon\) while \(v^ \varepsilon\) converges weakly-\(*\) to 0 in the appropriate topology as \(\varepsilon\) tends to zero. The field \(v^ \varepsilon\) contains a wealth of information which is lost in the limit process. In particular the energy density associated to \(v^ \varepsilon\), i.e., \[ d^ \varepsilon = {1\over 2} \left( \rho^ \varepsilon \left( {\partial v^ \varepsilon \over \partial t} \right)^ 2 + A^ \varepsilon \text{grad} v^ \varepsilon \text{grad} v^ \varepsilon \right) \] satisfies \(H^ \varepsilon {\buildrel {\text{def}} \over =} \int_ \Omega d^ \varepsilon (x,t) dx\) is independent of \(t\), and \(H^ \varepsilon @> \varepsilon \to 0>>H\), where \(H\) is a positive (and in general strictly positive) constant.
Our goal in the present paper is to further describe the local behaviour of \(d^ \varepsilon\) as \(\varepsilon\) tends to zero; for example we will aim at computing the measure limit \(d^ 0\) of \(d^ \varepsilon\). From a more physical standpoint we strive to understand the space-time localization properties of the part of the elastic energy that remains trapped during the homogenization process.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bensoussan A., Asymptotic Analysis for Periodic Structures (1978) · Zbl 0404.35001
[2] Brahim–Otsmane S., J. Maths. pures et appl. (1992)
[3] Deny J., Ann. Inst. Fourier 5 pp 305– (1955) · Zbl 0065.09903
[4] Folland G. B., Lectures on Partial Differential Equations, Tata Institute of Fundamental Research (1989) · Zbl 0671.58036
[5] P. Gérard Microlocal Defect Measures, Communications in PDE’s, volume in honor of R.J. Di Perna, to appear
[6] Gérard P., Seminaire Ecole Polytechnique (1988)
[7] Gérard P., Collège de France Seminar (1989)
[8] Hormander L., The analysis of linear partial differential operators (1985)
[9] Lax P. D., Lecture Notes, Stanford (1963)
[10] Maslov V. P., Math. Physics and Applied Maths. 7 (1981)
[11] Mizohata S., The theory of partial differential equations (1973) · Zbl 0263.35001
[12] Murat F., Publications Labo 6 (1976)
[13] Tartar L., Collég de France 1977, partially written by F. Murat in: Hndashconvergence, séminaire d’analyse fonctionnelle et numérique de l’Université d’Alger 6 (1977)
[14] Tartar L., Proc. Royal Soc. Ed. pp 193– (1977)
[15] Tartar L., Propagation of Oscillations and Concentration Effects in Partial Differential Equations, in Proceedings 11th Dundee Conference (1990) · Zbl 0774.35008
[16] Tartar L., H–measures and applications (1990)
[17] Treves F., Introduction to Pseudo–Differential Operators and Fourier Integral Operators 1 (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.