Vector field method for wave equations

The vector field method commutes a wave equation with spacetime symmetry generators, estimates the resulting commuted wave energies, and converts weighted L2 norms to pointwise decay using a Klainerman-Sobolev inequality. The commutation vector fields for the wave equation include translations, rotations, Lorentz boost vector fields and the scaling vector field. It is useful for almost global existence for wave equations and for small-data global existence under the classical null condition for wave equations.

A commuted wave energy controls derivatives of a solution after applying the commutation vector fields for the wave equation. A convenient equivalent norm is $A_m(t)={\sum }_{|I|\le m}\Vert \partial Z^{I}u(t){\Vert }_2$. The wave energy estimate controls its growth through commuted sources, while the Klainerman-Sobolev inequality gives pointwise decay for low-order derivatives.

In three spatial dimensions, with $Z$ the eleven translations, spatial rotations, boosts and scaling commutation vector fields for the wave equation, a sufficiently decaying smooth function satisfies $|u(t,x)|\le C(1+t+|x|{)}^{-1}(1+|t-|x||{)}^{-1/2}{\sum }_{|I|\le 2}\Vert Z^{I}u(t){\Vert }_2$. This weighted Sobolev inequality converts control of commuted wave energy into decay. It holds independently of any wave equation satisfied by $u$.

A first-order vector field $Z$ is a commutation field for the flat wave equation when $[\square ,Z]$ is zero or a controlled multiple of $\square$. Translations, spatial rotations and Lorentz boost vector fields commute with $\square$; the scaling vector field satisfies $[\square ,S]=2\square$. Repeated commutation preserves the differential order of a semilinear wave equation.

The vector field $S=t{\partial }_t+x\cdot \nabla$ generates simultaneous spacetime dilation. For the flat d'Alembert operator, $[\square ,S]=2\square$. Consequently it commutes with the homogeneous wave equation at the level of its solution set, although its operator commutator is nonzero.

For Minkowski spacetime with unit light speed, $L_i=t{\partial }_i+x_i{\partial }_t$ generates a Lorentz boost. It commutes with the d'Alembert operator and controls derivatives transverse to time slices in the vector field method for wave equations.

The vector field ${\Omega }_{ij}=x_i{\partial }_j-x_j{\partial }_i$ is an infinitesimal spatial rotation. It commutes with the flat d'Alembert operator and supplies angular derivatives in the Klainerman-Sobolev inequality.

The vector fields ${\partial }_t$ and ${\partial }_i$ generate time and space translations and commute with the flat d'Alembert operator. They form the unweighted part of the commutation vector fields for the wave equation.