Bilinear wave estimates

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Bilinear estimates

  • Let d > 1. If φ, ψ are free \dot H^{s_1} and \dot H^{s_2} solutions respectively, then one can control φψ in \dot X^{s,b} if and only if
    • (Scaling) s + b = s1 + s2 − (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

2 Q_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 φ, ψ are \dot H^{s- a } solutions, then
D^{-2 a } ( \phi\psi ) \in L^{q/2}_t L^{r/2}_x

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.

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