Program
 An introduction to Maxwell-Klein-Gordon equation Student No.： 50 Time： Tue/Thu 10:10-12:00, 2015-05-12 ~ 2015-05-28 Instructor： Shuang Miao  [University of Michigan] Place： Conference Room 3, Floor 2, Jin Chun Yuan West Building Starting Date： 2015-5-12 Ending Date： 2015-5-28

Description:

The Maxwell-Klein-Gordon (MKG) equation is the Maxwell equation coupled with a free wave equation or a Klein-Gordon equation. The coupling is through the covariant differentiation operator for the scalar field, which has curvature form equal to the Maxwell field. This coupling makes the equation nonlinear. If the scalar field is set to 0, the equation reduces to linear Maxwell equation, and if the Maxwell field is trivial, the equation for the scalar field reduces to the free wave (or Klein-Gordon) equation. The MKG equation models the charged scalar field and it is a special case of Yang-Mills-Higgs equation when the Lie group G is simply the unit circle U(1). In this mini course, we will first present some basic tools to study the MKG equation, which are also useful in many other problems, and then discuss some recent results on the global-in-time behavior of solutions to the MKG equation. The focus is on the structural aspect of the equation.

Prerequisite:

Basic real analysis and differential geometry

Reference:

(1) W-T. Shu. Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space. Comm. Math. Phys. 1991.

(2) H. Lindblad and J. Sterbenz. Global stability for charged-scalar fields on Minkowski space. IMRP Int. Math. Res. Pap. 2006.

(3) L. Bieri, S. Miao and S. Shahshahani. Asymptotic properties of solutions of the Maxwell Klein Gordon equation with small data. arxiv:1408.2550.