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arXiv 0912.4400math.AP
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三维空间中具有二次非线性项的wave equation

On the wave equation with quadratic nonlinearities in three space dimensions

Axel Gruenrock

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摘要：

考虑三维空间中非线性波动方程$$\Box u=(\partial u)^2, \qquad u(0)=u_0, u_t(0)=u_1$$的Cauchy问题.假设数据$(u_0,u_1)$属于$\widehat{H}^r_s(\R^3) \times \widehat{H}^r_{s-1}(\R^3)$，其中$\widehat{H}^r_s$由范数$$\n{f}{\widehat{H}^r_s} := \n{< ξ> ^s\widehat{f}}{L^{r'}_ξ},\quad < ξ>=(1+|ξ|^2)^{\frac12}, \quad \frac{1}{r}+\frac{1}{r'}=1.$$定义。局部适定性在参数范围$2 \ge r >1$，$s > 1 + \frac{2}{r}$中示出。对于$r=2$，这与Ponce和Sideris的结果一致，其通过Lindblad的反例在$H^s$尺度上是最优的，但是仍然留下了$\frac12$导数到尺度预测的差距。除了端点的情况之外，这个差距在这里是封闭的。对$\Box u = \partial u^2$也得到了相应的结果。

英文摘要：

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 $$\n{f}{\widehat{H}^r_s} := \n{< ξ> ^s\widehat{f}}{L^{r'}_ξ},\quad < ξ>=(1+|ξ|^2)^{\frac12}, \quad \frac{1}{r}+\frac{1}{r'}=1.$$ 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 $\frac12$ 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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