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### Hyperbolic systems with constant coefficients

We consider the -periodic constant coefficients system

Define the Fourier symbol associated with P(D):

which arises naturally when we Fourier transform (hyper.18),

Solving the ODE (hyper.20) we find, as before, that hyperbolicity amounts to

For this to be true the necessary condition should hold, namely

: For the wave equation, (1.1.4), .
But the Gårding-Petrovski condition is not sufficient for the hyperbolic estimate (1.1.18) as told by the counterexample

As before, in this case we have , hence the Gårding-Petrovski condition is fulfilled. Yet, Fourier analysis shows that we need both and in order to upperbound . Thus, the best we can hope for with this counterexample is an a priori estimate of the form

We note that in this case we have a "loss" of one derivative, and this brings us to the notion of

: We say that the system (1.1.17) is weakly hyperbolic if there exists an such that the following a priori estimate holds:

The Gårding-Petrovski condition is necessary and sufficient for the system (hyper.18) to be weakly hyperbolic. A necessary and sufficient characterization of hyperbolic systems is provided by the Kreiss matrix theorem: it states that (hyper.21) holds iff there exists a positive symmetrizer such that

and this yields the conservation of the -weighted norm, ; that is,

is conserved in time.

: For an a priori estimate forward in time (), it will suffice to have

Indeed, we have in this case

and hence summing over all k's and using Parseval's equality

Two important subclasses of hyperbolic equations are the strictly hyperbolic systems -- where has distinct real eigenvalues so that can be real diagonalized

and as before, will do; the other important case consists of symmetric hyperbolic systems which can be symmetrizer in the physical space, i.e. there exists an H > 0 such that

Most of the physically relevant systems fall into these categories.

: Shallow water equations (linearized)

with

can be symmetrized with

Next: Hyperbolic systems with variable Up: Initial Value Problems of Previous: The wave equation --