Difference between revisions of "Bilinear Airy estimates"

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== Algebraic identity ==
 
== Algebraic identity ==
  
Much of the bilinear estimate theory for [[Airy equation]] rests on the following ``three-wave resonance identity'':
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Much of the bilinear estimate theory for [[Airy equation]] rests on the following "three-wave resonance identity":
  
 
<center><math>\xi_1^3 + \xi_2^3 + \xi_3^3 = 3 \xi_1 \xi_2 \xi_3</math> whenever <math>\xi_1 + \xi_2 + \xi_3 = 0</math></center>
 
<center><math>\xi_1^3 + \xi_2^3 + \xi_3^3 = 3 \xi_1 \xi_2 \xi_3</math> whenever <math>\xi_1 + \xi_2 + \xi_3 = 0</math></center>
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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. [[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]].
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[[Category:Estimates]]

Revision as of 05:02, 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:

|| (uv)_x ||_{-3/4+, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}

|| (uv)_x ||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-3/8+, 1/2+}
|| (uv)_x ||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}
  • Remark: In principle, a complete list of bilinear estimates could be obtained from [[references.html#Ta-p2 Ta-p2]].