A polygonal scheme and the lower bound on density for the isentropic gas dynamics
A polygonal scheme and the lower bound on density for the isentropic gas dynamics
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DOI：
10.3934/dcds.2019172
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发表时间：
2019-04
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影响因子：
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通讯作者：
Geng Chen;R. Pan;Shengguo Zhu
中科院分区：
文献类型：
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作者：
Geng Chen;R. Pan;Shengguo Zhu
正密度下界是拉格朗日坐标下一维等熵可压缩欧拉方程(又称p-系统)大数据理论的主要障碍之一。Riemann首先研究的显式例子表明，即使当初始密度一致为正时，密度的下界也可以随着时间的推移而衰减到无穷大的顺序：Begin{Document}$O\Left({\frac{1}{{1+t}\Right)$\end{Document}。本文利用多边形格式的方法证明了密度的下界按其最优顺序为：开始{文档}$O\左({\frac{1}{{1+t}\右)$\end{文档}。
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 \begin{document}$ O\left( {\frac{1}{{1 + t}}} \right)$\end{document} , even when initial density is uniformly positive. In this paper, we establish a proof of the lower bound on density in its optimal order \begin{document}$ O\left( {\frac{1}{{1 + t}}} \right)$\end{document} using a method of polygonal scheme.