| In this dissertation I will study the asymptotic behavior of solutions of the initial value problem for the spin equations in Minkowski space and Schwarzchild space. Then I will prove the global existence and asymptotic behavior of solutions of Yang-Mills equations for small initial data. The main results are the three corollaries after Lemma 8, Theorem 3 and Theorem 4. The method is that of energy estimation. It can be generalized to prove the same results for those space times constructed by Christodoulou and Klainerman in (5); combined with their proof there, one can also prove the global existence of solution of the initial value problem for Einstein equation coupled with Yang-Mills equation. In S3, I will define the Lie derivative for spinors, and I will also discuss some of the properties of this Lie derivative operator. These properties are very natural and very simple to prove, and can be used to simplify many of the calculations in (5). |
