Wave estimates

Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms $L_{t}^{q}L_{x}^{r}$, or in $X_{}^{s,b}$ spaces, defined by

$\|u\|_{X^{s,b}}=\|<\xi >^{s}<|\xi |-|\tau |>^{b}{\hat {u}}(\tau ,\xi )\|_{2}$

Linear space-time estimates are known as Strichartz estimates. They are especially useful for the semilinear NLW without derivatives, and also have applications to other non-linearities, although the results obtained are often non-optimal (Strichartz estimates do not exploit any null structure of the equation). The $X_{}^{s,b}$ spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces first appear in one-dimension in RaRe1982 and in higher dimensions in Be1983 in the context of propagation of singularities; they were used implicitly for LWP in KlMa1993, while the Schrodinger and KdV analogues were developed in Bo1993, Bo1993b.

Specific wave estimates

• Linear wave estimates (Strichartz, etc.)
• Bilinear wave estimates
• Multilinear wave estimates