Fourier-integral operator approximation of solutions to first-order hyperbolic pseudodifferential equations II: microlocal analysis

Jérôme Le Rousseau (1) , Gunther Hormann (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 Fakultät für Mathematik [Wien]

An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolic first-order pseudodifferential equation, $\d_z + a(z,x,D_x)$ with $\Re (a) \geq 0$, is constructed as the composition of global Fourier integral operators with complex phases. We investigate the propagation of singularities for this Ansatz and prove microlocal convergence: the wavefront set of the approximated solution is shown to converge to that of the exact solution away from the region where the phase is complex.

Mots clés: Fourier integral operator with complex phase; microlocal analysis; convergence; infinite products; first order hyperbolic equation; pseudodifferential operator; Seismic imaging

Domaines: Equations aux dérivées partielles [math.AP]; Géophysique [physics.geo-ph]; Géophysique [physics.geo-ph]

Jérôme Le Rousseau, Gunther Hormann. Fourier-integral operator approximation of solutions to first-order hyperbolic pseudodifferential equations II: microlocal analysis. Journal de Mathématiques Pures et Appliquées, 2006, 86, pp.403-426. ⟨10.1016/j.matpur.2006.08.004⟩. ⟨hal-00005759⟩