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The wave equation -- hyperbolicity by Fourier analysis

Fourier transform (hyper.4b) to get the ODE
whose solution is
where tex2html_wrap_inline11007 is the Fourier transform of the initial data. Now, for
we find
put differently, we have
and hence (since the diagonal matrix inside the brackets on the right is clearly unitary), the tex2html_wrap_inline11009-norm of tex2html_wrap_inline11011 is conserved in time,i.e.,
Summing over all modes and using Parseval's equality we end up with energy conservation
as asserted.
We note that the only tool used in the energy method was the existence of a positive symmetrizer for A, while the only tool used in the Fourier method was the real diagonalization of A; in fact the two are related, for if tex2html_wrap_inline11017, then with tex2html_wrap_inline11019 we have

Energy conservation implies (in view of linearity) uniqueness, and serves as a basic tool to prove existence. It will be taken as the definition of hyperbolicity. It implies and is implied by the qualitative properties (1)--(4) which opened our discussion on page  gif.

We now turn to consider general PDE's of the form
with tex2html_wrap_inline11001-periodic boundary conditions and subject to prescribed initial conditions, u(x,0) = f(x) . Motivated by the example of the wave equation, we make the definition of

: We say that the system (1.1.17) is hyperbolic if the following a priori energy estimate holds:

As we shall see later on, this notion of hyperbolicity is equivalent with energy conservation ( -- measured with respect to an appropriate renormed weighted 'energy'), in analogy with what we have seen in the special case of the wave equation. Here are the basic facts concerning such systems.

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