Bilinear wave estimates: Difference between revisions

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===Bilinear estimates===
===Bilinear estimates===


* Let <math>d>1</math> . If <math>f</math>, <math>y</math> are free <math>\dot H^{s_1}</math> and <math>\dot H^{s_2}</math> solutions respectively, then one can control fy  in \dot X^{s,b} if and only if
* Let <math>d>1</math>. If <math>\phi</math>, <math>\psi</math> are free <math>\dot H^{s_1}</math> and <math>\dot H^{s_2}</math> solutions respectively, then one can control <math>\phi\psi</math> in <math>\dot X^{s,b}</math> if and only if
** (Scaling) <math>s+b = s_1 + s_2 - (d-1)/2</math>
** (Scaling) <math>s+b = s_1 + s_2 - (d-1)/2</math>
** (Parallel interactions) <math>b \geq (3-d)/4</math>
** (Parallel interactions) <math>b \geq (3-d)/4</math>
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** (No double endpoints) <math>(s_1, b), (s_2, b) \neq ((d+1)/4, -(d-3)/4); (s_1+s_2, b) \neq (1/2, -(d-3)/4)</math>.
** (No double endpoints) <math>(s_1, b), (s_2, b) \neq ((d+1)/4, -(d-3)/4); (s_1+s_2, b) \neq (1/2, -(d-3)/4)</math>.


See [[Bibliography#FcKl2000|FcKl2000]]. Null forms can also be handled by identities such as
See [[FcKl2000]]. Null forms can also be handled by identities such as


<center><math>2 Q_0( f , y ) = \Box( f  y ).</math></center>
<center><math>2 Q_0( \phi , \psi ) = \Box( \phi, \psi ).</math></center>


* Some bilinear Strichartz estimates are also known. For instance, if <math>s</math>, <math>q</math>, <math>r</math> are as in the linear Strichartz estimates  <math>f</math> , <math>y</math> are <math>\dot H^{s- a }</math> solutions, then
* Some bilinear [[Strichartz estimate]]s are also known. For instance, if <math>s</math>, <math>q</math>, <math>r</math> are as in the linear Strichartz estimates  <math>\phi</math>, <math>\psi</math> are <math>\dot H^{s- a }</math> solutions, then


<center><math>D^{-2 a } ( fy ) is in L^{q/2}_t L^{r/2}_x</math></center>
<center><math>D^{-2 a } ( \phi\psi ) \in L^{q/2}_t L^{r/2}_x</math></center>


as long as <math>0 \leq  a  \leq d/2 - 2/q - d/r</math> [[Bibliography#FcKl-p |FcKl-p]]. Similar estimates for null forms also exist [[Bibliography#Pl2002|Pl2002]]; see also [[Bibliography#TaVa2000b|TaVa2000b]], [Ta-p4].
as long as <math>0 \leq  a  \leq d/2 - 2/q - d/r</math> [[FcKl2000]]. Similar estimates for null forms also exist [[Pl2002]]; see also [[TaVa2000b]], [[Ta2001b]].


[[Category:Wave]]
[[Category:Estimates]]  
[[Category:Estimates]]
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Latest revision as of 20:10, 4 March 2007

Bilinear estimates

  • Let . If , are free and solutions respectively, then one can control in if and only if
    • (Scaling)
    • (Parallel interactions)
    • (Lack of smoothing)
    • (Frequency cancellation)
    • (No double endpoints) .

See FcKl2000. Null forms can also be handled by identities such as

  • Some bilinear Strichartz estimates are also known. For instance, if , , are as in the linear Strichartz estimates , are solutions, then

as long as FcKl2000. Similar estimates for null forms also exist Pl2002; see also TaVa2000b, Ta2001b.

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