A polygonal scheme and the lower bound on density for the isentropic gas dynamics

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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\left( {\frac{1}{{1 + t}}} \right)$, 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\left( {\frac{1}{{1 + t}}} \right)$ using a method of polygonal scheme.