I-method

The I-method is a method for constructing global solutions to nonlinear dispersive and wave equations in situations when the relevant conserved quantity (such as the energy E(u)) is subcritical but infinite. One applies a mollifying operator I to the solution (dependent on a large frequency truncation parameter N) to make the conserved quantity E(Iu) finite. The catch is that this quantity is no longer conserved, but one hopes to show an almost conservation law for this quantity which makes it stable over long periods of time (going to infinity as $N\to \infty$). Letting N go to infinity one obtains global well-posedness.

The I-method was inspired by the earlier Fourier truncation method of Bourgain, which achieves similar effects by approximating the evolution of the rough solution by the evolution if its low and high frequency components.