Formation of Singularities in One-Dimensional Hydromagnetic Flow**The project supported by National Natural Science Foundation of China under Grant No. 10001024, the Special Funds for Major State Basic Research Projects No. G2000077306, the Qi Ming Xing programme of Shanghai Government, and the Project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China

Abstract

Two results on the formation of singularities in solutions to the system of one-dimensional hydromagnetic dynamics are presented. In particular, it is shown that shocks form from a smooth spatial periodic flow in a finite time if the initial amounts of entropy and the “magnetic field” in each period are smaller than those of sound waves. A quantitative estimate of blow-up time is also given.