For the heat equation

\begin{equation} u_t(t,x) = \nu u_{xx}(t,x) \end{equation}

for $x \in [0,1]$ with boundary conditions $u(t,0) = u(t,1) = 0$ and initial value $u(0,x) = u_0(x)$ it is easy to show that the "energy" defined as

\begin{equation} E(t) = \frac{1}{2} \int_{0}^{1} u^2(t,x)~dx \end{equation}

decays over time, that is $E(t) \leq E(0)$. I wonder if there is any physical interpretation of the quantity $E$?

In terms of units, $u$ is temperature in Kelvin while the thermal diffusivity $\nu$ has units $m^2/s$ and is composed from \begin{equation} \nu = \frac{k}{\rho c_p} \end{equation} where $k$ is the thermal conductivity in $W/(m*K)$, $\rho$ the density in $kg/m^3$ and $c_p$ the specific heat capacity in $J/(kg*K)$.

I figured out that $u$ is related to the internal energy in Joule per volume via \begin{equation} I = c_p \rho u. \end{equation}

But what is the interpretation of $E$, which is related to $u^2$?

  • 2
    $\begingroup$ You've stated the differential equation without specifying boundary conditions (or the initial condition for that matter). The boundary conditions play a role in proving the energy decay inequality you ask about. $\endgroup$
    – hardmath
    Oct 27 '16 at 14:27
  • $\begingroup$ I added the BC and initial data for the sake of completeness, but I fail to see how they will help the interpretation of $E(t)$? $\endgroup$
    – Daniel
    Oct 28 '16 at 7:14
  • $\begingroup$ $E$ has no physical meaning for the heat equation. it is just a mathematical construct - you prove solution decay in an "energy" norm. for the wave equation $u_{tt} = c^2 \Delta u$, however, we have a similar argument based on the total wave energy (potential + kinetic) as described, e.g., here $\endgroup$
    – GoHokies
    Oct 28 '16 at 18:12

In these slide there are some comments about the energy.

At pag 4 it focus on the fact that this energy is not a physical energy, but it is a mathematical tool.

At pag 8 it observes that:

From a physical point of view it seems reasonable that a the energy will decrease in a system without any heat source

And after this the author defines another kind of energy based on $u_x$. Also this energy is decreasing (with the same boundary conditions).

  • $\begingroup$ Hi Mauro, the usefulness of the "energy" as a mathematical concept is clear, but I was hoping that there was a physical interpretation, too. But it seems this is not the case. Thanks for the slides! $\endgroup$
    – Daniel
    Oct 29 '16 at 9:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.