On defining appropriate energy. Any principle? I am reading Evans' book Partial differential equations. but I am really curious about how he define the appropriate energy? Is there any principle or rule to do this things? Because I notice that although the energies he defined in this book are always contain kinetic energy and potential energy, they are always contain other terms. I mean the details are really different. For example, in Chapter 12 of his book on nonlinear wave equations. He defined the energy of equation 
$$
u_{tt}-\Delta u+f(u)=0
$$
as
$$
E(t):=\int_{R^n}\frac{1}{2} (u_t^2+|Du|^2)+F(u) dx 
$$
where
$$
F(z):=\int_{0}^{z}f(w)dw 
$$
However, He also define the energy of equation
$$
u_{tt}-\Delta u+f(Du,u_t,u)=0
$$ 
as
$$
E(t):=\frac{1}{2}\int_{R^n} u_t^2+|Du|^2+u^2 dx
$$
I wonder why he didn't define the same energy (might not get the what we want?), or I  should say why he defined the second energy just like the first one or vice versa.
Hoping your answers. Many thanks. 
 A: Physical interpretation can be very helpful, but ultimately we need $E$ to be such that we can 


*

*infer something about $dE/dt$ from the PDE 

*infer what we want about the solution from $E$


The consideration usually begins with 1. Note that $u_tu_{tt} = \frac{1}{2}(u_t)^2$. So, if we have a hyperbolic PDE  $u_{tt}=F(x,u,Du,D^2u)$, it is reasonable to multiply both sides by $u_t$ and try to see if a part of the right hand-side can also be turned into the $t$-derivative of  something, possibly after integration by parts in the $x$ variable. This works for the Laplacian:
$$
\int u_t \Delta u  = - \int D_x(u_t)\, D_xu = \frac{d}{dt} \int \frac12 |Du|^2
$$
Also works for anything of the form $f(u)$, because $u_t f(u)= \frac{\partial }{\partial t} F(u)$ by the chain rule, if $F$ is an antiderivative of $f$. 
However, the term $f(Du,u_t,u)$ is not so simple. A superficial reason is that there is no "antiderivative" $F$, as $f$ is a function of three variables. More substantial reason (again from physics) is that $f$ is a force that depends on derivatives $Du$ and $u_t$, and such a force is not conservative — it does not have an energy potential. 
Evans includes $u^2$ in the short-time existence proof for the nonlinear wave equation (Theorem 3 in Chapter 12) not because it makes for a neat identity/inequality for $dE/dt$ (there isn't one) but because $\int u^2$ also needs to be estimated to prove the theorem (see item 2 above).
