Fourier transform (hyper.4b) to get the ODE
whose solution is
where is the Fourier transform of the initial data. Now, for
put differently, we have
and hence (since the diagonal matrix inside the brackets on the right is clearly unitary), the -norm of is conserved in time,i.e.,
Summing over all modes and using Parseval's equality we end up with energy conservation
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 , then with 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 .
We now turn
to consider general PDE's of the form
with -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
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.