On the wave equation with quadratic nonlinearities in three space dimensions

Abstract

The Cauchy problem for the nonlinear wave equation $$\square u=(\partial u{)}^2,u(0)=u_0,u_t(0)=u_1$$ in three space dimensions is considered. The data $(u_0,u_1)$ are assumed to belong to ${\hat{H}}_s^r({\mathbb{R}}^3)\times {\hat{H}}_{s-1}^r({\mathbb{R}}^3)$, where ${\hat{H}}_s^r$ is defined by the norm $$\n{f}{\widehat{H}^r_s} := \n{< ξ> ^s\widehat{f}}{L^{r'}_ξ},\quad < ξ>=(1+|ξ|^2)^{\frac12}, \quad \frac{1}{r}+\frac{1}{r'}=1.$$ 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 $\frac{1}{2}$ derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for $\square u=\partial u^2$ are obtained, too.

References in corpus (2)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• On the hierarchies of higher order mKdV and KdV equations