Sur le système hyperbolique à coefficients constants

Canonical Forms of 3 x 3 Strongly and Nonstrictly Hyperbolic Systems with Complex Constant Coefficients

“…It was Yamaguti and Kasahara [4], [9] who gave the definition and a criterion for the system (1.1) to be strongly hyperbolic (see Theorem 2.4 below for their criterion). Later, Strang [8] proved that (1.1) is strongly hyperbolic if and only if its initial value problem is L 2 -wellposed.…”

“…We usually denote real constants by lower case greek letters, complex constants by lower case roman letters, matrices by upper case roman letters unless otherwise indicated. is said to be the matrix family spanned by A l9 A 2 , ..., A n and is denoted by <,A l9 A 29 ... 9 A H y. Definition 23o A matrix family <A l9 A 2 , ..., A n y is called uniformly realdiagonalizable if it is real-diagonalizable and there is a diagonalizer S(£) such that there exists a constant M > 0 independent of £ for which when £, runs over R n .…”

“…Theorem 2A (Yamaguti-Kasahara [9]). Equation (1.1) is strongly hyperbolic if and only if the matrix family (A l9 A 2 , ..., A n y is uniformly realdiagonalizable.…”

“…As mentioned in Introduction, strong hyperbolicity is equivalent to L 2 -wellposedness in the case of constant coefficients. For the proof of Theorem 2.4, see Yamaguti-Kasahara [9], Kasahara-Yamaguti [4] (B°°-theory) 3 or Strang [8] (L 2 -theory).…”

“…A matrix family (A l9 A 2 ,..., A n y is said to have multiple eigenvalues if some A(£) e (A l9 A 2 ,..., A n y with £ ^ (0,..., 0) does. In addition, a strongly hyperbolic system (1.1) is said to be strongly and nonstrictly hyperbolic if ^A i9 A 29 ... 9 A n y has multiple eigenvalues.…”