I-method

From DispersiveWiki


The I-method (or the method of almost conserved quantities) 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 ). Letting N go to infinity one obtains global well-posedness.

One can extend the reach of the I-method by adding correction terms to the quantity E(Iu) to make it oscillate more slowly, though this has so far proven unable to extend the global existence results all the way down to the critical regularity.

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.

Explanation for the name I

The name I was chosen for the mollification operator, because this operator acts like the Identity on low freqeuencies (less than N), and like an Integration operator on high frequencies. The I-team used this operator extensively in their joint papers, hence the name (though the I was also interpreted to indicate the International nature of the collaboration).