The Small Data Solutions of General 3D Quasilinear Wave Equations. I

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For the three-dimensional (3D) quasilinear wave equation $\sum_{i,j=0}^3g^{ij}(\partial u)\partial_{ij}^2u=0$ with coefficients independent of the solution $u$, a blowup result or global existence result has been completely established provided that the null condition does not hold or holds, respectively. In this paper and a subsequent paper (see [D. Bingbing, W. Ingo, and Y. Huicheng, The Small Data Solutions of General $3$-D Quasilinear Wave Equations. II, preprint, 2014]), we will systematically study the more general 3D quasilinear wave equation $\sum_{i,j=0}^3g^{ij}(u, \partial u)\partial_{ij}^2u=0$ with coefficients depending simultaneously on $u$ and $\partial u$. When the weak null condition holds, we show that the smooth small data solution exists globally in the paper.