The block structure condition for symmetric hyperbolic systems

In the analysis of hyperbolic boundary value problems, the construction of Kreiss’ symmetrizers relies on a suitable block structure decomposition of the symbol of the system. In this paper, we show that this block structure condition is satisfied by all symmetrizable hyperbolic systems of constant multiplicity. In [2], H.O.Kreiss proved a maximal L2 energy estimate for the solutions of mixed boundary-initial value problems for strictly hyperbolic systems and boundary conditions which satisfy the uniform Lopatinski condition (see also [8] for systems with complex coefficients). The proof is based on the construction of a symmetrizer. Thanks to the pseudodifferential calculus, the proof is reduced to the construction of an algebraic symmetrizer for the symbol of the equation (see e.g. [1]). The result extends to the case where the coefficients have finite smoothness ([3]) and, using the paradifferential calculus of J.M.Bony-Y.Meyer, to Lipschitzean coefficients ([7],[6]). Kreiss’ analysis is extended to a class of characteristic boundary value problems in [5]. However, many interesting physical examples of hyperbolic systems are not strictly hyperbolic. For instance, Euler’s equations of gas dynamics, Maxwell’s equations or the equations of elasticity are not strictly hyperbolic. In the construction of Kreiss’ symmetrizer, the strict hyperbolicity assumption is used at only one place, to prove that the symbol of the system has a suitable block decomposition near glancing modes (see Lemmas 2.5, 2.6 and 2.7 in [2]). In [5] and [3], it is shown that this block structure condition is satisfied by several nonstrictly hyperbolic systems such as the linearized shock front equations of gas dynamics ([3]), Maxwell’s equations or the linearized shallow water equations ([5]). However, due to the lack of a simple criterion, one had to check the condition for each system separately. The aim of this paper is to prove that the block structure assumption is satisfied for a large class of systems of physical interest which contains the examples above : the class of symmetrizable hyperbolic systems of constant multiplicity. As a corollary, continuing the analysis as in [2], [1], [5] or [3], this implies the local well posedness of boundary value problems for linear symmetric (or