Fourier-integral-operator approximation of solutions to first-order hyperbolic pseudodifferential equations I: convergence in Sobolev spaces
Jerome Le Rousseau
TL;DR
We study the Cauchy problem for the first-order hyperbolic pseudodifferential equation $\partial_z u + a(z,x,D_x)u = 0$ with ${\Re}(a) \ge 0$, and construct an approximate operator solution $U(z',z)$ as a composition of global Fourier integral operators with complex phases. The thin-slab propagator $\mathcal{G}_{(z',z)}$ is introduced with complex phase $\phi_{(z',z)}$ and amplitude; it maps Sobolev spaces and satisfies global Sobolev bounds, in particular $\|\mathcal{G}_{(z',z)}\|_{(s,s)} \le 1+M\Delta$ for small thickness $\Delta = z'-z$. A subdivision-based approximation $\mathcal{W}_{\mathfrak{P},z}$, built from composing thin-slab propagators, converges to the true solution operator $U(z,0)$ in $L(H^{(s+1)},H^{(s)})$ with rate $\Delta_{\mathfrak{P}}^{1/2}$ under Lipschitz continuity of $a$ in $z$, and a Kumano-go-type variant $\widehat{\mathcal{W}}_{\mathfrak{P},z}$ attains the same rate under analogous hypotheses. These results provide a rigorous operator-approximation framework for hyperbolic pseudodifferential equations via Fourier integral operators, enabling controlled approximations of propagation in variable media, with microlocal analysis to be addressed in Part II.
Abstract
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 $\infty$.