标题： 二维外区域二次拟线性波动方程的全局光滑解，第二部分 显示英文标题

标题： Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, II

摘要： 在论文[S. Alinhac, 二维空间中二次拟线性波动方程的零条件 I, Invent. Math. 145 (2001), no. 3, 597-618]中，S. Alinhac建立了带有零条件的二维二次拟线性波动方程柯西问题的小数据光滑解的全局存在性。然而，对于外区域中的相应二维初边值问题，全局解是否存在仍然是一个开放问题。当二维二次非线性具有特殊的$Q_0$类型零形式时，我们在之前的论文[Hou Fei, Yin Huicheng, Yuan Meng, 外区域中带有零条件的二维二次拟线性波动方程的全局光滑解, arXiv:2411.06984]中展示了全局小解的存在性。在本文中，我们现在通过证明外区域中带有零条件的二维一般拟线性波动方程的小解的全局存在性来解决这个开放问题。我们的证明过程基于在零条件下找到拟线性波动方程适当的散度结构，引入一个良好的未知量以消除产生的$Q_0$类型非线性，并推导出解及其导数的一些新的精确点态时空衰减估计。 显示英文摘要

摘要： In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy problem of 2-D quadratically quasilinear wave equations with null conditions. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. When the 2-D quadratic nonlinearity admits a special $Q_0$ type null form, the global small solution is shown in our previous article [Hou Fei, Yin Huicheng, Yuan Meng, Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, arXiv:2411.06984]. In the present paper, we now solve this open problem through proving the global existence of small solutions to 2-D general quasilinear wave equations with null conditions in exterior domains. Our proof procedure is based on finding appropriate divergence structures of quasilinear wave equations under null conditions, introducing a good unknown to eliminate the resulting $Q_0$ type nonlinearity and deriving some new precise pointwise spacetime decay estimates of solutions and their derivatives.