Much of the trilinear estimate theory for Airy equation rests on (various permutations of) the following "four-wave resonance identity":
- The key algebraic fact is (various permutations of)
The following trilinear estimates are known:
The 1/4 is sharp KnPoVe1996.We also have
- The 1/2 estimate CoKeStTkTa-p3 on T: if have mean zero, then
The 1/2 is sharp KnPoVe1996.
- Remark: the trilinear estimate always needs one more derivative of regularity than the bilinear estimate; this is consistent with the heuristics from the Miura transform from mKdV to KdV.