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:
- How do you know what the "appropriate" definition of energy is for a given PDE?
- Does every PDE have a useful well-defined notion of energy?
- 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