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)

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

Estimates

The following trilinear estimates are known:

• The 1/4 estimate references.html#Ta-p2 Ta-p2 on R:

|| (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 references.html#KnPoVe1996 KnPoVe1996.We also have

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

see [Cv-p].

• The 1/2 estimate references.html#CoKeStTaTk-p3 CoKeStTkTa-p3 on T: if u,v,w have mean zero, then

|| (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 references.html#KnPoVe1996 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.