Let the transport equation in $\mathbb{R}^n:$: $$\frac{\partial u}{\partial t}(x,t) + c. \nabla u(x,t) = 0 $$
Where $x \in \mathbb{R}^n$ and $t>0$. $u$ is a regular function with compact support. The energy of the solution is defined $$ E(t) = \frac{1}{2}\int_{\mathbb{R}^n}u^2(x,t)dx$$
By computing $\frac{d E}{dt}$ prove that the energy is conserved.
My approach was this one, but I struggle to finish it. To show that the energy is conserved, we have to show that $E$ is constant, thus $\frac{dE}{dt}$ is equal to zero.
I thus have:
$$\frac{dE}{dt} = \frac{dE}{dt} \frac{1}{2}\int_{\mathbb{R}^n}u^2(x,t)dx = \frac{1}{2}\int_{\mathbb{R}^n}\frac{d}{dt}u^2(x,t)dx$$
$$\frac{2}{2}\int_{\mathbb{R}^n}(\frac{d}{dt}u(x,t)) u(x,t)dx $$
As we have $\frac{\partial u}{\partial t}(x,t) + c. \nabla u(x,t) = 0$ we thus have $\frac{\partial u}{\partial t}(x,t) = -c . \nabla u(x,t)$
Thus we have:
$$\int_{\mathbb{R}^n}(\frac{d}{dt}u(x,t)) u(x,t)dx = \int_{\mathbb{R}^n}-c . \nabla u(x,t) \cdot u(x,t)dx $$
And here I don't know how to proceed.
Best Answer
Your last term can be evaluated by the following observation:
$$ \nabla u\cdot u = u_1\frac{\partial u_1}{\partial x_1} +\ldots + u_1\frac{\partial u_n}{\partial x_n}. $$
Over $\mathbb{R}^n$ each of these can be evaluated as
$$ \lim_{a\to\infty}\int_{-a}^a u_i \frac{\partial u_i}{\partial x_i}dx_i = \lim_{a\to\infty} \left . \frac{1}{2} u_i^2 \right |_{-a}^a = 0. $$