Fourier-integral-operator approximation of solutions to first-order hyperbolic pseudodifferential equations I: convergence in Sobolev spaces

Jérôme Le Rousseau (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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. An estimate of the operator norm in $L(H^{(s)},H^{(s)})$ of these operators is provided which allows to prove a convergence result for the Ansatz to $U(z',z)$ in some Sobolev space as the number of operators in the composition goes to $\infty$.

Mots clés: Fourier integral operator with complex phase; microlocal analysis; Sobolev regularity; 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. Fourier-integral-operator approximation of solutions to first-order hyperbolic pseudodifferential equations I: convergence in Sobolev spaces. Communications in Partial Differential Equations, 2006, 31, pp.867-906. ⟨10.1080/03605300600635079⟩. ⟨hal-00003815v3⟩