Note on strongly hyperbolic systems with involutive characteristics

Abstract

We consider the Cauchy problem in $L^2$ for first-order systems. A necessary condition is that the system must be uniformly diagonalizable or, equivalently, that it admits a bounded symmetrizer. A sufficient condition is that it admits a smooth (Lipschitz) symmetrizer, which is true when the system is diagonalizable with eigenvalues of constant multiplicities. Counterexamples show that uniform diagonalizability is not sufficient in general for systems with variable coefficients, and they indicate that the symplectic properties of the set $\Sigma$ of the singular points of the characteristic variety are important. In this article, we give a new class of systems for which the Cauchy problem is well-posed in $L^2$. The main assumption is that $\Sigma$ is a smooth involutive manifold and the system is transversally strictly hyperbolic.