Difference between revisions of "Bilinear Airy estimates"

From DispersiveWiki
(Estimates)
(Estimates)
Line 24: Line 24:
 
<center><math>\| u \partial_x v\|_{X^{s, -1/2}} \lesssim \| u \|_{X^{s, 1/2}} \| v \|_{X^{s, 1/2}}</math></center>
 
<center><math>\| u \partial_x v\|_{X^{s, -1/2}} \lesssim \| u \|_{X^{s, 1/2}} \| v \|_{X^{s, 1/2}}</math></center>
  
** The above estimate fails for <math>s < -1/2</math>. Also, one cannot replace <math>1/2, -1/2 by 1/2+, -1/2+<math>. [[references.html#KnPoVe1996 KnPoVe1996]]
+
** The above estimate fails for <math>s < -1/2</math>. Also, one cannot replace <math>1/2, -1/2 by 1/2+, -1/2+</math>. [[references.html#KnPoVe1996 KnPoVe1996]]
 
** This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]
 
** This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]
 
* ''Remark''<nowiki>: In principle, a complete list of bilinear estimates could be obtained from [</nowiki>[references.html#Ta-p2 Ta-p2]].
 
* ''Remark''<nowiki>: In principle, a complete list of bilinear estimates could be obtained from [</nowiki>[references.html#Ta-p2 Ta-p2]].
  
 
[[Category:Estimates]]
 
[[Category:Estimates]]

Revision as of 15:13, 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:


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