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Hyperbolic systems with constant coefficients

We consider the tex2html_wrap_inline11001-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), tex2html_wrap_inline11029.
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 tex2html_wrap_inline11029, hence the Gårding-Petrovski condition is fulfilled. Yet, Fourier analysis shows that we need both tex2html_wrap_inline11033 and tex2html_wrap_inline11035 in order to upperbound tex2html_wrap_inline11037. 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 tex2html_wrap_inline11039 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 tex2html_wrap_inline11041 such that
and this yields the conservation of the tex2html_wrap_inline11009-weighted norm, tex2html_wrap_inline11045; that is,
is conserved in time.

: For an a priori estimate forward in time (tex2html_wrap_inline11047), 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 tex2html_wrap_inline11051 has distinct real eigenvalues so that tex2html_wrap_inline11051 can be real diagonalized
and as before, tex2html_wrap_inline11055 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)
can be symmetrized with

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

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998