On the wave equation with quadratic nonlinearities in three space dimensions

The Cauchy problem for the nonlinear wave equation $$\mathrm{\square }u=(\mathrm{\partial }u{)}^2,u(0)=u_0,u_t(0)=u_1$$ in three space dimensions is considered. The data $(u_0,u_1)$ are assumed to belong to $\widehat{H}^r_s(\R^3) \times \widehat{H}^r_{s-1}(\R^3)$,where$\widehat{H}^r_s$isdefinedbythenorm$$\n{f}{\widehat{H}^r_s} := \n{< \xi > ^s\widehat{f}}{L^{r’}_{\xi}},\quad < \xi

=(1+|\xi|^2)^{\frac12}, \quad \frac{1}{r}+\frac{1}{r’}=1.$$ Local well-posedness is shown in the parameter range $2\ge r>1$, $s > 1 + \frac{2}{r}$\mathrm{.}For$r=2$ this coincides with the result of Ponce and Sideris, which is optimal on the $H^s$-scale by Lindblad’s counterexamples, but nonetheless leaves a gap of $\frac{1}{2}$ derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for $\mathrm{\square }u=\mathrm{\partial }{u}^2$ are obtained, too.