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 [[ | 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> [[ | 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].
