On the wave equation with quadratic nonlinearities in three space dimensions

TL;DR

This paper establishes local well-posedness for the three-dimensional quadratic nonlinear wave equation in a broad function space range, improving previous results and nearly matching the scaling prediction, with some exceptions.

Contribution

It extends the well-posedness results for the quadratic wave equation in three dimensions to a larger function space range, nearly reaching the scaling limit.

Findings

Well-posedness proven for 2 > r > 1 and s > 1 + 2/r

Results match optimal known results for r=2, close to scaling prediction

Includes results for both f u^2 and f (f u)^2 equations.

Abstract

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.