Fourier-integral-operator product representation of solutions to first-order symmetrizable hyperbolic systems
Résumé
We consider the first-order Cauchy problem \begin{align*} \partial_z u + a(z,x,D_x) u &=0, \ \ \ 0< z\leq Z,\\ u \mid_{z=0} &= u_0, \end{align*} with $Z>0$ and $a(z,x, D_x)$ a $k\times k$ matrix of pseudodifferential operators of order one, whose principal part is assumed symmetrizable: there exists $L(z,x,\xi)$ of order $0$, invertible, such that \begin{align*} a_1 (z,x,\xi) = L(z,x,\xi)\; (- i \beta_1(z,x,\xi) + \gamma_1(z,x,\xi))\; (L(z,x,\xi))^{-1}, \end{align*} where $\beta_1$ and $\gamma_1$ are hermitian symmetric and $\gamma_1\geq 0$. An approximation Ansatz for the operator solution, $U(z',z)$, is constructed as the composition of global Fourier integral operators with complex matrix phases. In the symmetric case, an estimate of the Sobolev operator norm in $L((H^{(s)}(\R^n))^k,(H^{(s)}(\R^n))^k)$ of these operators is provided, which yields a convergence result for the Ansatz to $U(z',z)$ in some Sobolev space as the number of operators in the composition goes to $\infty$, in both the symmetric and symmetrizable cases. We thus obtain a representation of the solution operator $U(z',z)$ as an infinite product of Fourier integral operators with matrix phases.
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