HAL : hal-00194089, version 2

DOI : 10.1007/s11854-009-0029-9

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J. Analyse Math. 109 (2009) 81-162

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v1 (05-12-2007)

v2 (29-01-2009)

Fourier-integral-operator product representation of solutions to first-order symmetrizable hyperbolic systems

Jérôme Le Rousseau 1, 2

(2009)

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.

1 :	Laboratoire d'Analyse, Topologie, Probabilités (LATP)
	CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III
2 :	Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO)
	Université d'Orléans – CNRS : UMR7349

Domaine

:

Mathématiques/Equations aux dérivées partielles

Mots Clés : Hyperbolic system – Symmetrizable system – Pseudodifferential initial value problem – Fourier integral operator – Matrix phase function – Global Sobolev norm estimate – Multi-product

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Contributeur : Jérôme Le Rousseau <>

Soumis le : Jeudi 29 Janvier 2009, 09:29:23

Dernière modification le : Mardi 16 Mars 2010, 17:21:16