High Frequency Limit of the Helmholtz EquationsReport as inadecuate

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1 ONDES - Modeling, analysis and simulation of wave propagation phenomena Inria Paris-Rocquencourt, Univ. Paris-Saclay, ENSTA ParisTech - École Nationale Supérieure de Techniques Avancées, CNRS - Centre National de la Recherche Scientifique : UMR2706 2 M3N - Multi-Models and Numerical Methods Inria Paris-Rocquencourt

Abstract : We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term which does not share the quadratic aspect in the limit, then, the lack of L2 bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity.

Author: Jean-David Benamou - François Castella Thodoros Katsaounis Benoît Perthame -

Source: https://hal.archives-ouvertes.fr/


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