Bilinear wave estimates: Difference between revisions

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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( f ,  y ) = \Box( f  y ).</math></center>
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<center><math>D^{-2 a } ( fy ) is in L^{q/2}_t L^{r/2}_x</math></center>
<center><math>D^{-2 a } ( fy ) is 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> [[FcKl-p]]. Similar estimates for null forms also exist [[Pl2002]]; see also [[TaVa2000b]], [Ta-p4].


   
   
[[Category:Estimates]]  
[[Category:Estimates]]  
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Revision as of 14:33, 10 August 2006

Bilinear estimates

  • Let . If , are free and solutions respectively, then one can control fy in \dot X^{s,b} 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 FcKl-p. Similar estimates for null forms also exist Pl2002; see also TaVa2000b, [Ta-p4].