Periodic entropy weak solutions for quasi–one–dimensional isentropic Euler flows with periodic initial data

Entropy solutions with periodic initial data are considered for the quasi-one-dimensional isentropic Euler system. An approximate solution sequence is constructed using a fractional Glimm scheme, and then wave interactions are carefully analyzed. Energy behavior of the system is studied as an extra approximate conservation law in order to overcome the difficulty caused by the non–conservation property. After that, the method of approximate conservation laws and approximate characteristics is applied to analyze the decay of the system, which gives the uniform total variation bounds and thus the global existence.