Difference between revisions of "Bilinear Airy estimates"

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The following bilinear estimates are known:
 
The following bilinear estimates are known:
  
* The <math>-3/4+</math> estimate [[Bibliography#KnPoVe1996|KnPoVe1996]] on '''R'''<nowiki>:</nowiki>
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* The <math>-3/4+</math> estimate [[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 <math> -3/4</math>. [[Bibliography#NaTkTs2001|NaTkTs2001]]
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** The above estimate fails at the endpoint <math> -3/4</math>. [[NaTkTs2001]]
** As a corollary of this estimate we have the -3/8+ estimate [[Bibliography#CoStTk1999|CoStTk1999]] on '''R'''<nowiki>: If u and v have no low frequencies ( |\xi| <~ 1 ) then</nowiki>
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** As a corollary of this estimate we have the -3/8+ estimate [[CoStTk1999]] on '''R'''<nowiki>: If u and v have no low frequencies ( |\xi| <~ 1 ) then</nowiki>
  
 
<center><math>\| u \partial_x v \|_{X^{{0, -1/2+}}} \lesssim \| u \|_{X^{-3/8+, 1/2+}} \| v \|_{X^{-3/8+, 1/2+}}</math></center>
 
<center><math>\| u \partial_x v \|_{X^{{0, -1/2+}}} \lesssim \| u \|_{X^{-3/8+, 1/2+}} \| v \|_{X^{-3/8+, 1/2+}}</math></center>
  
* The -1/2 estimate [[Bibliography#KnPoVe1996|KnPoVe1996]] on '''T'''<nowiki>: if </nowiki><span class="SpellE">u,v</span> have mean zero, then for all s >= -1/2
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* The -1/2 estimate [[KnPoVe1996]] on '''T''': if u,v have mean zero, then for all <math>s \geq -1/2</math>
  
 
<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>. [[Bibliography#KnPoVe1996|KnPoVe1996]]
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** The above estimate fails for <math>s < -1/2</math>. Also, one cannot replace <math>1/2, -1/2</math> by <math>1/2+, -1/2+</math>. [[KnPoVe1996]]
** This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. [[Bibliography#CoKeStTkTa-p2 |CoKeStTkTa-p2]]
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** This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. [[CoKeStTkTa2003]]
* ''Remark''<nowiki>: In principle, a complete list of bilinear estimates could be obtained from </nowiki>[[Bibliography#Ta-p2 |Ta-p2]].
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* ''Remark'': In principle, a complete list of bilinear estimates could be obtained from [[Ta2001]].
  
 
[[Category:Estimates]]
 
[[Category:Estimates]]

Latest revision as of 19:36, 4 March 2007

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:

  • The estimate KnPoVe1996 on R:


    • The above estimate fails at the endpoint . NaTkTs2001
    • As a corollary of this estimate we have the -3/8+ estimate CoStTk1999 on R: If u and v have no low frequencies ( |\xi| <~ 1 ) then
  • The -1/2 estimate KnPoVe1996 on T: if u,v have mean zero, then for all
    • The above estimate fails for . Also, one cannot replace by . KnPoVe1996
    • This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. CoKeStTkTa2003
  • Remark: In principle, a complete list of bilinear estimates could be obtained from Ta2001.