Microlocal Energy Density for Hyperbolic Systems
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.
KeywordsWave Equation Hyperbolic System Pseudodifferential Operator Dual Variable Scientific Computing
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