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.

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Thanks for your post. Since FT is an isometry with -norm, so it’s very nice on the conservation of total energy .

Comment by Tuan Minh — March 24, 2010 @ 17:26

Actually I have lots of estimates regarding to wave and heat equations (KdV, NST, etc.), hopefully I have enough time to show.

Comment by Ngô Quốc Anh — March 24, 2010 @ 17:33

Since the total energy only depends on the initial data (Schwartz functions) so we can imply the uniqueness of the solution to the initial value problem for the wave equation in the upper half-plane. The condition that f,g are Schwartz functions is too strong. Indeed, you can see the exercise (3, 5D) in the well-known book of Folland that we only need , then

Comment by Tuan Minh — March 7, 2012 @ 18:59