A POLYGONAL SCHEME AND THE LOWER BOUND ON DENSITY FOR THE ISENTROPIC GAS DYNAMICS

摘要

Positive density lower bound is one of the major obstacles toward large data theory for one dimensional isentropic compressible Euler equations, also known as p-system in Lagrangian coordinates. The explicit example first studied by Riemann shows that the lower bound of density can decay to zero as time goes to infinity of the order O(1/1+t), even when initial density is uniformly positive. In this paper, we establish a proof of the lower bound on density in its optimal order O(1/1+t) using a method of polygonal scheme.