# Difference between revisions of "Bilinear Airy estimates"

From DispersiveWiki

Jump to navigationJump to searchLine 1: | Line 1: | ||

== Algebraic identity == | == Algebraic identity == | ||

− | Much of the bilinear estimate theory for [[Airy equation]] rests on the following | + | 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 27: | Line 27: | ||

** 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]] |

## 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":

## Estimates

The following bilinear estimates are known:

- The -3/4+ estimate references.html#KnPoVe1996 KnPoVe1996 on
**R**:

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

- 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**: If u and v have no low frequencies ( |\xi| <~ 1 ) then

`|| (uv)_x ||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-3/8+, 1/2+}`

- The -1/2 estimate references.html#KnPoVe1996 KnPoVe1996 on
**T**: if u,v have mean zero, then for all s >= -1/2

`|| (uv)_x ||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}`

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