Applied Mathematics and Scientific Computing pp 179-190 | Cite as

# Microlocal Energy Density for Hyperbolic Systems

Chapter

## Abstract

Starting from the method for computing microlocal energy density, which was developed independently by Francfort and Murat, and Gérard for the linear wave equation, we compute that very density for the hyperbolic system

We express the energy limit for the sequence of initial problems in terms of the energy of initial conditions. The basic tool we use are H-measures (also known as microlocal defect measures). We associate an H-measure to the sequence of gradients of solutions to our system and it represents the desired microlocal energy density.

We determine the system of equations satisfied by the corresponding H-measure. In the case of constant coefficients it reduces to a hyperbolic system similar to the initial one. Finally, we give a few examples related to the wave equation.

## Keywords

Wave Equation Hyperbolic System Pseudodifferential Operator Dual Variable Scientific Computing
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© Springer Science+Business Media New York 2002