Bilinear wave estimates

Bilinear estimates

• Let $d>1$. If $\phi$, $\psi$ are free ${\dot {H}}^{s_{1}}$ and ${\dot {H}}^{s_{2}}$ solutions respectively, then one can control $\phi \psi$ in ${\dot {X}}^{s,b}$ if and only if
  • (Scaling) $s+b=s_{1}+s_{2}-(d-1)/2$
  • (Parallel interactions) $b\geq (3-d)/4$
  • (Lack of smoothing) $s\leq s_{1},s_{2}$
  • (Frequency cancellation) $s_{1}+s_{2}\geq 1/2$
  • (No double endpoints) $(s_{1},b),(s_{2},b)\neq ((d+1)/4,-(d-3)/4);(s_{1}+s_{2},b)\neq (1/2,-(d-3)/4)$.

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

$2Q_{0}(\phi ,\psi )=\Box (\phi ,\psi ).$

• Some bilinear Strichartz estimates are also known. For instance, if $s$, $q$, $r$ are as in the linear Strichartz estimates $\phi$, $\psi$ are ${\dot {H}}^{s-a}$ solutions, then

$D^{-2a}(\phi \psi )\in L_{t}^{q/2}L_{x}^{r/2}$

as long as $0\leq a\leq d/2-2/q-d/r$ FcKl2000. Similar estimates for null forms also exist Pl2002; see also TaVa2000b, Ta2001b.