Difference between revisions of "Bilinear Airy estimates"

Algebraic identity

Much of the bilinear estimate theory for Airy equation rests on the following "three-wave resonance identity":

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

Estimates

The following bilinear estimates are known:

$\displaystyle \| u\partial_x v ||_{X^{-3/4+, -1/2+}} \lesssim \| u \|_{X^{{-3/4+, 1/2+}} \| v \|_{X^{{-3/4+, 1/2+}}$

${\displaystyle \|u\partial _{x}v\|_{X^{0,-1/2+}}\lesssim \|u\|_{X^{-3/8+,1/2+}}\|v\|_{X^{-3/8+,1/2+}}}$
${\displaystyle \|u\partial _{x}v\|_{X^{s,-1/2}}\lesssim \|u\|_{X^{s,1/2}}\|v\|_{X^{s,1/2}}}$
• Remark: In principle, a complete list of bilinear estimates could be obtained from [[references.html#Ta-p2 Ta-p2]].