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

Thu Jan 22 19:07:34 PST 1998