Instructor: Zhiwu Lin

Description:We will study stability of various relative equlibria in nonlinear waves, fluid and plasmas. The nonlinear wave equations include Scrodinger, Klein-Gorden and KDV type equations from water wave theory. Their stability theory has became rather completed and we will give a summary of this theory including some recent refinements. The fluid and plasma stability is much less systematic and many advances has only been obtained in recent years. We will pick the following typical models to illustrate some general issues and methods: 2D Euler equation for ideal plane flows with a fixed or free boundary as in water waves, Vlasov-Poisson systems for electrostatic plasmas and galaxy dynamics, and Vlasov-Maxwell systems for collisionless plasmas. We will study the linearized stability and instability, and how to pass the linear theory to the nonlinear level. In particular, we will consider the imporatnt role played by particle dynamics in recent studies of plasma and fluid stability.

Text: Many material are from recent papers. For backgrounds on the PDEs models, we refer to Walter Strauss, nonlinear wave equations, 1989.
Robert Glasey, Cauchy problems in kinetic theory, 1996. Carlo Marchioro and Mario Pulvirenti, Mathematical theory of incompressible nonviscous fluids, 1994.