Trilinear Airy estimates: Difference between revisions

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The following trilinear estimates are known:
The following trilinear estimates are known:


* The 1/4 estimate [[references.html#Ta-p2 Ta-p2]] on '''R'''<nowiki>:</nowiki>
* The 1/4 estimate [[Bibliography#Ta-p2 |Ta-p2]] on '''R'''<nowiki>:</nowiki>


<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uvw</span><span class="GramE">)_</span>x ||_{1/4, -1/2+} <~ || u ||_{1/4, 1/2+} || v ||_{1/4, 1/2+} || w ||_{1/4, 1/2+}</font></tt></center>
<center><math>
|| (uvw)_x ||_{1/4, -1/2+} <~ || u ||_{1/4, 1/2+} || v ||_{1/4, 1/2+} || w ||_{1/4, 1/2+}
</math></center>


The 1/4 is sharp [[references.html#KnPoVe1996 KnPoVe1996]].We also have
The 1/4 is sharp [[Bibliography#KnPoVe1996 |KnPoVe1996]].We also have


<center><tt><font size="10.0pt"><nowiki>|| </nowiki><span class="SpellE"><span class="GramE">uv<u>w</u></span></span> ||_{-1/4, -5/12+} <~ || u ||_{-1/4, 7/12+} || v ||_{-1/4, 7/12+} || w ||_{-1/4, 7/12+}</font></tt></center>
<center><math>|| uvw ||_{-1/4, -5/12+} <~ || u ||_{-1/4, 7/12+} || v ||_{-1/4, 7/12+} || w ||_{-1/4, 7/12+}
</math></center>


<span class="GramE">see</span> [<span class="SpellE">Cv</span>-p].
<span class="GramE">see</span> [<span class="SpellE">Cv</span>-p].


* The 1/2 estimate [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]] on '''T'''<nowiki>: if </nowiki><span class="SpellE">u,v,w</span> have mean zero, then
* The 1/2 estimate [[Bibliography#CoKeStTkTa-p3 |CoKeStTkTa-p3]] on '''T'''<nowiki>: if </nowiki><span class="SpellE">u,v,w</span> have mean zero, then


<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uvw</span><span class="GramE">)_</span>x ||_{1/2, -1/2} <~ || u ||_{1/2, 1/2*} || v ||_{1/2, 1/2*} || w ||_{1/2, 1/2*}</font></tt></center>
<center><math>|| (uvw)_x ||_{1/2, -1/2}  
<~ || u ||_{1/2, 1/2*} || v ||_{1/2, 1/2*} || w ||_{1/2, 1/2*}
</math></center>


The 1/2 is sharp [[references.html#KnPoVe1996 KnPoVe1996]].
The 1/2 is sharp [[Bibliography#KnPoVe1996 |KnPoVe1996]].


* ''Remark''<nowiki>: the </nowiki><span class="SpellE">trilinear</span> estimate always needs one more derivative of regularity than the bilinear estimate; this is consistent with the heuristics from the Miura transform from <span class="SpellE">mKdV</span> to <span class="SpellE">KdV</span>.
* ''Remark''<nowiki>: the </nowiki><span class="SpellE">trilinear</span> estimate always needs one more derivative of regularity than the bilinear estimate; this is consistent with the heuristics from the Miura transform from <span class="SpellE">mKdV</span> to <span class="SpellE">KdV</span>.


[[Category:Estimates]]
[[Category:Estimates]]

Revision as of 16:37, 31 July 2006

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)
whenever

Estimates

The following trilinear estimates are known:

  • The 1/4 estimate Ta-p2 on R:

The 1/4 is sharp KnPoVe1996.We also have

see [Cv-p].

  • The 1/2 estimate CoKeStTkTa-p3 on T: if u,v,w 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.