On the wave equation with quadratic nonlinearities in three space dimensions

Details On the wave equation with quadratic nonlinearities in three space dimensions On the wave equation with quadratic nonlinearities in three space dimensions

2009-12-22

arxiv.org/abs/0912.4400v1

Mathematics

Analysis of PDEs

Scientific paper

The Cauchy problem for the nonlinear wave equation $$\Box u=(\partial u)^2, \qquad u(0)=u_0, u_t(0)=u_1$$ 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 $\widehat{H}^r_s$ is defined by the norm $$\n{f}{\widehat{H}^r_s} := \n{< \xi > ^s\widehat{f}}{L^{r'}_{\xi}},\quad < \xi >=(1+|\xi|^2)^{\frac12}, \quad \frac{1}{r}+\frac{1}{r'}=1.$$ Local well-posedness is shown in the parameter range $2 \ge r >1$, $s > 1 + \frac{2}{r}$. 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 $\frac12$ derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for $\Box u = \partial u^2$ are obtained, too.

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