A Gaussian beam approach for computing Wigner measures in convex domains

TL;DR

This paper introduces a Gaussian beam method to analyze high-frequency wave energy in convex domains, utilizing Wigner measures to track energy transport along broken bicharacteristics, with explicit computation and generalization to standard initial data.

Contribution

It presents a novel Gaussian beam approach combined with Wigner measures for high-frequency wave analysis in convex domains, including explicit computation and generalization techniques.

Findings

Explicit computation of Wigner measures for Gaussian beams.

Validation of energy transport along broken bicharacteristics.

Extension of results to standard initial conditions.

Abstract

A Gaussian beam method is presented for the analysis of the energy of the high frequency solution to the mixed problem of the scalar wave equation in an open and convex subset, with initial conditions compactly supported in this set, and Dirichlet or Neumann type boundary condition. The transport of the microlocal energy density along the broken bicharacteristic flow at the high frequency limit is proved through the use of Wigner measures. Our approach consists first in computing explicitly the Wigner measures under an additional control of the initial data allowing to approach the solution by a superposition of first order Gaussian beams. The results are then generalized to standard initial conditions.