弯曲背景下非线性wave equation的共形散射
Conformal scattering for a nonlinear wave equation on a curved background
摘要：
本文的目的是建立渐近简单时空上共形不变非线性波动方程的几何散射结果。散射算子是通过零无穷远处的迹算子得到的。证明分三步完成。先验的线性估计通过适应的Morawetz向量场在史瓦西时空和Hörmander使用的Goursat问题的方法。然后得到了无穷远处光锥上特征柯西问题的适定性结果。这就需要控制的非线性均匀的时间，这来自一个估计的Sobolev常数和衰减假设的非线性方程。最后，构造了共形无穷远域上的迹算子，并利用迹算子定义了共形散射算子。
英文摘要：
The purpose of this paper is to establish a geometric scattering result for a conformally invariant nonlinear wave equation on an asymptotically simple spacetime. The scattering operator is obtained via trace operators at null infinities. The proof is achieved in three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field in the Schwarzschild spacetime and a method used by Hörmander for the Goursat problem. A well-posedness result for the characteristic Cauchy problem on a light cone at infinity is then obtained. This requires a control of the nonlinearity uniform in time which comes from an estimates of the Sobolev constant and a decay assumption on the nonlinearity of the equation. Finally, the trace operators on conformal infinities are built and used to define the conformal scattering operator.
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