Local regularity of non-resonant nonlinear wave equations

Abstract

We study the problem of minimal regularity required to ensure local well-posedness for systems of nonlinear wave equations with different propagation speeds in three space dimensions $$ \begin{align} (\partial_t^2 -C_1^2\Delta)u &= F(u,v,\partial u,\partial v), \\ (\partial_t^2 -C_2^2\Delta)v &= G(u,v,\partial u,\partial v). \end{align} $$ We prove that if $C_2 > C_1$ and $F$, $G$ have the form $\p u \cdot v$, then the problem is well-posed in $H^1$. Our proof is based on the same type of space-time estimates as those of Klainerman and Machedon.