# Physical interpretation of L2 norm of heat equation solution

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 $u^2$?

• 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. – hardmath Oct 27 '16 at 14:27
• I added the BC and initial data for the sake of completeness, but I fail to see how they will help the interpretation of $E(t)$? – Daniel Oct 28 '16 at 7:14
• $E$ 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 – GoHokies Oct 28 '16 at 18:12

## 1 Answer

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).

• 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! – Daniel Oct 29 '16 at 9:05