We prove global existence backwards from the scattering data posed at infinity for the Maxwell Klein Gordon equations in Lorenz gauge satisfying the weak null condition. The asymptotics of the solutions to the Maxwell Klein Gordon equations in Lorenz gauge were shown to be wave like at null infinity and homogeneous towards timelike infinity in Candy et al. (Commun Math Phys 367(2):683–716, 2019) and expressed in terms of radiation fields, and thus our scattering data will be given in the form of radiation fields in the backward problem. We give a refinement of the asymptotics results in Candy et al. (2019), and then making use of this refinement, we find a global solution which attains the prescribed scattering data at infinity. Our work starts from the approach in [21] and is more delicate since it involves with nonlinearities with fewer derivatives. Our result corresponds to “existence of scattering states” in the scattering theory. The method of proof relies on a suitable construction of the approximate solution from the scattering data, a weighted conformal Morawetz energy estimate and a spacetime version of Hardy inequality.

我们从满足弱零条件的Lorenz规范中的Maxwell Klein Gordon方程的无穷大散射数据向后证明了整体存在。在Candy等人的研究中，Lorenz量表的Maxwell Klein Gordon方程解的渐近性表现为波状，在零无穷大处，并且向时间无穷大均匀。（Commun Math Phys 367（2）：683–716，2019），并以辐射场表示，因此，在向后问题中，我们的散射数据将以辐射场的形式给出。我们对Candy等人的渐近结果进行了细化。（2019），然后利用这种改进，我们找到了一个全局解决方案，该解决方案在无穷远处获得了规定的散射数据。我们的工作从[21]中的方法开始，并且更加精细，因为它涉及具有较少导数的非线性。我们的结果对应于散射理论中的“散射状态的存在”。证明方法依赖于散射数据，加权共形Morawetz能量估计和Hardy不等式的时空版本的近似解的适当构造。