三维拟线性波方程的小初值光滑解
Small Data Solutions of 3-D Quasilinear Wave Equations

对三维小初值拟线性波方程$\sum\limits_{i,j=0}^3g^{ij}( u)\p_{ij}u=0$, H. Lindblad 证明了它有整体光滑解. 本文考虑如下带有小初值的拟线性波方程$\sum\limits_{i,j=0}^3g^{ij}(u)\p_{ij} u=(\p u)^3$, 通过得到低阶导数的衰减估计和高阶导数的能量估计, 由连续论证法证明了这个方程也存在整体光滑解.
For the 3-D quasilinear wave equation $\sum\limits_{i,j=0}^3g^{ij}(u)\p_{ij}u=0$, a global existence result has been shown by H. Lindblad. This paper deals with this 3-D quasilinear wave equation $\sum\limits_{i,j=0}^3g^{ij}(u)\p_{ij} u=(\p u)^3$ with small initial data. Through deriving decay estimates of low derivatives and energy estimates for high derivatives, combined with known weighted energy inequality, a global existence solution is also established by continuous induction.