Trilinear Airy estimates

Algebraic identity

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)
${\displaystyle \xi _{1}^{3}+\xi _{2}^{3}+\xi _{3}^{3}+\xi _{4}^{3}=3(\xi _{1}+\xi _{4})(\xi _{2}+\xi _{4})(\xi _{3}+\xi _{4})}$ whenever ${\displaystyle \xi _{1}+\xi _{2}+\xi _{3}+\xi _{4}=0}$

Estimates

The following trilinear estimates are known:

${\displaystyle ||(uvw)_{x}||_{1/4,-1/2+}<~||u||_{1/4,1/2+}||v||_{1/4,1/2+}||w||_{1/4,1/2+}}$

The 1/4 is sharp KnPoVe1996.We also have

${\displaystyle ||uvw||_{-1/4,-5/12+}<~||u||_{-1/4,7/12+}||v||_{-1/4,7/12+}||w||_{-1/4,7/12+};}$

see Cv2004.

• The 1/2 estimate CoKeStTkTa-p3 on T: if ${\displaystyle u,v,w}$ have mean zero, then
${\displaystyle ||(uvw)_{x}||_{1/2,-1/2}<~||u||_{1/2,1/2*}||v||_{1/2,1/2*}||w||_{1/2,1/2*}}$

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.