Energy method

Summary

The energy method is a way to prove uniqueness, stability and a priori bounds for solutions of partial differential equations. One attaches to a solution a nonnegative quantity, often a physical energy, and shows from the equation that it cannot grow too fast. For evolution equations this gives control of the solution at time $t$ by its data at time $0$, which is the core of well-posedness. It works for many hyperbolic and parabolic equations where no explicit formula is available.