A few questions about energy methods for PDEs

Question

I took a full-year undergrad course on PDE's a few years ago, and now I'm a grad student (in pure math) taking a grad course on PDE's. While PDE's aren't my main interest, I'm sufficiently interested that I've spent the intervening time thinking about/reading about PDE's as well. There are a few things about energy methods that I haven't been able to find a clear answer to (or, at least, an answer that makes sense to me), and it's really starting to gnaw at me.

I guess everything can be summed up in the following questions:

1. How do you know what the "appropriate" definition of energy is for a given PDE?
2. Does every PDE have a useful well-defined notion of energy?
3. If I were studying a new PDE for which there wasn't much/any general theory, is there a generally accepted line of inquiry that would lead me to the appropriate notion of energy?

Where I'm currently at:

• I understand how the definition of energy came about for the Laplace equation and the variational energy functional for more general Euler-Lagrange equations
• I don't have a feeling for why the energies of the heat or wave equations are "right"
• I don't know the associated energies of any other PDE's (if they even have any)
• I don't have a background in physics, and I'm assuming that the lack of physical intuition is at least part of the problem here

Answer 1

I think to pick up some physics background would be helpful. This does not have to be advanced physics--the wave equation for example applies to string vibration, which only requires Newton's F=ma. It also applies to electromagnetic waves which requires Maxwell's equations from 200 years after Newton. Many PDE come from some physical application, so carry an energy concept, but this does not mean that the only useful "energy" is the physical one. For example, in heat conduction along a metal rod $u_t = u_{xx}$ (modulo physical constants, more on that below) the energy content is $\int_0^L u(x,t) dx$ and the time derivative of that, using the heat equation and integration by parts, is $u_x(L,t)-u_x(0,t)$, which is the rate that heat conducts away at the ends. Very physical. But it is also useful to consider the time derivative of $\int_0^L u^2(x,t)dx$, because a uniqueness argument can be based on that one, taking $u$ to be the difference of two presumed solutions having the same initial and boundary conditions. If you want to see elementary discussions of some of these things that include all the physics constants I can suggest my Notes on Differential Equations here.