system with the low-frequency (and hence smooth) data in the energy space. To obtain a solution to the system with the full initial data, one solves a difference equation for the remainder, with the high-frequency initial data. To iterate, it must be proved that the nonlinear part of the solution to the difference equation is in the energy space (i.e. smoother than the initial data) with small norm. One can then add this smoother part to the low-frequency solution and repeat the process. If one can obtain strong enough bounds, any time interval $[0,T]$ can be covered via iteration, by choosing $N$ large enough.

The I-method works by applying a smoothing operator, again with some cut-off parameter $N$, to the system. A local theory is then obtained for this smoothed version of the system in the energy space. The next step is to show that a modified energy corresponding to the smoothed system is “almost conserved", i.e. it doesn’t not grow too rapidly. If the growth is slow enough, one can again iterate to cover any interval $[0,T]$ by choosing a sufficiently large cut-off value $N$.

In general, the I-method will yield well-posedness at lower Sobolev regularities than the high-low 