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...
Let us introduce the energy method on an example (see e.g. the book Partial Differential Equations by L.C. Evans for others). We consider the initial-value problem of the advection equation $$ u_t+u_x = 0 , \qquad u(x,0) = \phi(x) $$ where $\phi$ is smooth and $(x,t)\in(\Bbb R, \Bbb R_+)$. The unique strong solution is $u_0(x,t) = \phi(x-t)$. We prove uniqueness using the energy $E(u) = \int_{\Bbb R} u^2\text dx$, which is constant over time and positive. Now, let us assume that there is another solution $u_1(x,t)$. At $t=0$, we have $E(u_0-u_1) = 0$. Therefore, for all $t$, $E(u_0-u_1) = 0$. Since the energy $E(u)$ of a function $u$ is zero if and only if the function is zero, we have proven the uniqueness of the solution.
In general, we try to find a positive function of the variables which is decreasing or constant over time. For many systems, the definition of an energy follows from the physics, which makes things easier (no particular need for an intuition). In some cases, one can construct relevant energies by applying the principle of least action, in the framework of calculus of variations.