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January  2008, 20(1): 53-79. doi: 10.3934/dcds.2008.20.53

Weak-convergence methods for Hamiltonian multiscale problems

Alexander Mielke 1,

1. 

Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin

Received  April 2007 Revised  August 2007 Published  October 2007

We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $\Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $\Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $L^\infty$ coefficients.
Keywords: infinite-dimensional Hamiltonian and Lagrangian, wave equation, Gamma convergence, Homogenization, recovery operators., oscillator chain, effective Hamiltonian.
Mathematics Subject Classification: Primary: 35B27, 37Kxx; Secondary: 70Hxx, 74Q1.
Citation: Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete & Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53
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