Let us consider the following problem
Assume . Note that the Schwartz space consists of all indefinitely differentiable functions such that
for every multi-index and . We shall prove that the total energy at time
is constant in time.
Our approach is based on the Fourier transform. Note that the Fourier transform of a Schwartz function is defined by
It is worth noticing that the Fourier transform maps to itself. Taking the Fourier transform with respect to space variable we have
Solving the above initial problem of the ordinary differential equation, we get
As a consequence,
We are now in a position to apply Plancherel’s Theorem to get the desired result.
Theorem (Plancherel). Suppose . Then
Let us denote by this common value of the total energy. One can show that
The key point is to use the Riemann-Lebesgue lemma from harmonic analysis. We refer the reader to a book entitled Fourier Analysis due to Stein and Shakarchi.