Let us consider the following problem

with

.

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

and

.

Solving the above initial problem of the ordinary differential equation, we get

and therefore

.

As a consequence,

.

We are now in a position to apply Plancherel’s Theorem to get the desired result.

Theorem(Plancherel). Suppose . Then.

Moreover

.

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.