Intuition behind using energy estimate to prove existence and uniqueness of solution for Hyperbolic equation I am trying to understand the intution behind use of energy estimate to prove existence and uniqueness(which is clear the energy estimate) of solution to hyperbolic equations. What is the basic idea behind construction of energy density function?
Thanks in advance...
 A: A prototypical example: the wave equation (for sake of simplicity let's work in 1d, but the idea is general):
$$ \partial_t^2u=\partial_x^2u. $$
Let $v=\partial_tu$ and $w=\partial_xu$. Then the above equation becomes
$$ \partial_tv=\partial_xw. $$
Being two different partial derivative of the same function $u$, $v$ and $w$ are related by the interchangeability of partial derivatives:
$$ \partial_tw=\partial_xv. $$
The above two equations can be written as one in the matrix form
$$ \partial_t{v\choose w}=S\partial_x{v\choose w},$$
where $S$ is the symmetric matrix
$$ \begin{pmatrix} 0 & 1\\1 & 0 \end{pmatrix}. $$
On the other hand, under the standard bilinear product (let's work with real-valued function for simplicity)
$$\langle f,g \rangle=\int f(x)g(x)dx,$$
$\partial_x$ is an antisymmetric operator thanks to integration by parts:
$$\langle \partial_xf,g \rangle=\int \partial_xf(x)\cdot g(x)dx=-\int f(x)\partial_xg(x)dx=-\langle f,\partial_xg \rangle.$$
Since $S$ and $\partial_x$ commute, $S\partial_x$ is also antisymmetric:
$$ (S\partial_x)^T=\partial_x^T\circ S^T=-\partial_x\circ S=-S\partial_x. $$
Therefore the wave equation, in the variable $X=(v,w)^T$, looks like
$$ \partial_tX=AX, $$
where $A=S\partial_x$ is antisymmetric. In finite dimensional (bi-)linear algebra we know that the flow generated by an antisymmetric matrix preserves the bilinear form. This generalizes to the setting of infinite dimensional function spaces. Therefore the above evolution equation preserves the bilinear form
$$ \langle X,X \rangle=\int (v(x)^2+w(x)^2)dx=\int (\partial_tu(x)^2+\partial_xu(x)^2)dx. $$
The energy density pops out naturally:
$$ e(x)=\partial_tu(x)^2+\partial_xu(x)^2. $$
Voila! 

PS The above procedure can be generalized vastly to symmetric (or even symmetrizable) hyperbolic systems. For a textbook reference see Chapter 5 of Michael Taylor, Pseudodifferential Operators and Nonlinear PDE.
