Induction on energy

The induction on energy method, pioneered by Bourgain, seeks to control the size of solution at a given energy by assuming inductively that solutions at measurably smaller energies are already controlled in size. The point of doing this is that one can then reduce (via an approximate principle of superposition) to the case of solutions which are "irreducible" in that they cannot be decomposed as a superposition of two weakly interacting components of strictly smaller energy.

These arguments combine well with concentration compactness methods.