Bilinear wave estimates: Difference between revisions
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===Bilinear estimates=== | ===Bilinear estimates=== | ||
* 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 \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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<center><math>D^{-2 a } ( \phi\psi ) \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> [[FcKl2000]]. Similar estimates for null forms also exist [[Pl2002]]; see also [[TaVa2000b]], [[ | 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:Wave]] | ||
[[Category:Estimates]] | [[Category:Estimates]] | ||
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.
