Beijing Institute of Mathematical Sciences and Applications

This course explores the mathematical theory of shock formation and development in relativistic fluids, focusing on spherically symmetric solutions. Building on Christodoulou's foundational work in shock formation, the course examines the shock development problem—a free boundary problem where classical solutions to the Euler equations break down, and discontinuities (shocks) emerge. We begin with foundational concepts in relativistic fluid dynamics, including conservation laws, jump conditions, and acoustic geometry. The course then turns to the geometric description of shock formation, the maximal development of initial data, and the solution to the shock development problem—a free boundary problem with singular initial data. Through iterative methods and characteristic coordinates, we construct and analyze solutions beyond shock formation, emphasizing uniqueness, regularity, and physical implications.