Difference between revisions of "Bilinear Airy estimates"

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(Estimates)
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The following bilinear estimates are known:
 
The following bilinear estimates are known:
  
* The -3/4+ estimate [[references.html#KnPoVe1996 KnPoVe1996]] on '''R'''<nowiki>:</nowiki>
+
* The <math>-3/4+</math> estimate [[references.html#KnPoVe1996 KnPoVe1996]] on '''R'''<nowiki>:</nowiki>
  
 
<center><math>\| u\partial_x v ||_{X^{-3/4+, -1/2+}} \lesssim \| u \|_{X^{{-3/4+, 1/2+}} \| v \|_{X^{{-3/4+, 1/2+}}</math></center>
 
<center><math>\| u\partial_x v ||_{X^{-3/4+, -1/2+}} \lesssim \| u \|_{X^{{-3/4+, 1/2+}} \| v \|_{X^{{-3/4+, 1/2+}}</math></center>
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** The above estimate fails at the endpoint -3/4. [[references.html#NaTkTs-p NaTkTs2001]]
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** The above estimate fails at the endpoint <math> -3/4</math>. [[references.html#NaTkTs-p NaTkTs2001]]
 
** As a corollary of this estimate we have the -3/8+ estimate [[references.html#CoStTk1999 CoStTk1999]] on '''R'''<nowiki>: If u and v have no low frequencies ( |\xi| <~ 1 ) then</nowiki>
 
** As a corollary of this estimate we have the -3/8+ estimate [[references.html#CoStTk1999 CoStTk1999]] on '''R'''<nowiki>: If u and v have no low frequencies ( |\xi| <~ 1 ) then</nowiki>
  

Revision as of 15:11, 28 July 2006

Algebraic identity

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

whenever

Estimates

The following bilinear estimates are known:

Failed to parse (syntax error): {\displaystyle \| u\partial_x v ||_{X^{-3/4+, -1/2+}} \lesssim \| u \|_{X^{{-3/4+, 1/2+}} \| v \|_{X^{{-3/4+, 1/2+}}}


  • Remark: In principle, a complete list of bilinear estimates could be obtained from [[references.html#Ta-p2 Ta-p2]].