Difference between revisions of "Bilinear Airy estimates"

From DispersiveWiki
m (Estimates)
 
(9 intermediate revisions by 3 users not shown)
Line 1: Line 1:
 
== Algebraic identity ==
 
== Algebraic identity ==
  
Much of the bilinear estimate theory for [[Airy equation]] rests on the following ``three-wave resonance identity'':
+
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>
Line 9: Line 9:
 
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 [[KnPoVe1996]] on '''R'''<nowiki>:</nowiki>
  
<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uv</span><span class="GramE">)_</span>x ||_{-3/4+, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}</font></tt></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>
  
<math>\| u v \|_{-3/4+, -1/2+} <~ \| u \|_{-3/4+, 1/2+} \| v \|_{-3/4+, 1/2+}</math>
 
  
** The above estimate fails at the endpoint -3/4. [[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>
 
  
<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uv</span><span class="GramE">)_</span>x ||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-3/8+, 1/2+}</font></tt></center>
+
** The above estimate fails at the endpoint <math> -3/4</math>. [[NaTkTs2001]]
 +
** 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>
  
* The -1/2 estimate [[references.html#KnPoVe1996 KnPoVe1996]] on '''T'''<nowiki>: if </nowiki><span class="SpellE">u,v</span> have mean zero, then for all s >= -1/2
+
<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><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uv</span><span class="GramE">)_</span>x ||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}</font></tt></center>
+
* The -1/2 estimate [[KnPoVe1996]] on '''T''': if u,v have mean zero, then for all <math>s \geq -1/2</math>
  
** The above estimate fails for s < -1/2. Also, one cannot replace 1/2, -1/2 by 1/2+, -1/2+. [[references.html#KnPoVe1996 KnPoVe1996]]
+
<center><math>\| u \partial_x v\|_{X^{s, -1/2}} \lesssim \| u \|_{X^{s, 1/2}} \| v \|_{X^{s, 1/2}}</math></center>
** 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]].
+
** 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. [[CoKeStTkTa2003]]
 +
* ''Remark'': In principle, a complete list of bilinear estimates could be obtained from [[Ta2001]].
 +
 
 +
[[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.