In this paper, we study the rarefaction wave case of the regularized Riemann problem proposed by Chu, Hong and Lee in SIMA MMS, 2020, for compressible Euler equations with a small parameter \(\nu \). The solutions \(\rho _\nu \) and \(v_\nu \) of such problems stand for the density and velocity of gas flow near vacuum, respectively. We show that as \(\nu \) approaches 0, the solutions \(\rho _\nu \) and \(v_\nu \) converge to the solutions \(\rho \) and v, respectively, of pressureless compressible Euler equations in \(L^1\) sense. In addition, the \(L^1\) convergence rates of these physical quantities in terms of \(\nu \) are also investigated. The \(L^1\) convergences and convergence rates are proved by two facts. One is to invent an a priori estimate coupled with the iteration method to the high-order derivatives of Riemann invariants so that we obtain the uniform boundedness of \(\partial _{x}^{i} \rho _{\nu }\) (\(i=0,1,2\)) and \(\partial _{x}^{j} v_{\nu }\) (\(j=0,1,2,3\)) on the requisite regions. The other is about convexity of characteristic curves, which is used to estimate the distances among characteristic curves in terms of \(\nu \). These theoretic results are also supported by numerical simulations.

中文翻译：

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真空附近可压缩Euler方程的$$ L ^ {1} $$ L1收敛和近似解的收敛速度

在本文中，我们研究了Chu，Hong和Lee在SIMA MMS，2020中针对具有小参数\（\ nu \）的可压缩Euler方程提出的正则化Riemann问题的稀疏波动情形。这些问题的解\（\ rho _ \ nu \）和\（v_ \ nu \）分别代表真空附近气流的密度和速度。我们证明，随着\（\ nu \）接近0，解\（\ rho _ \ nu \）和\（v_ \ nu \）分别收敛到无压可压缩的解\（\ rho \）和v。\（L ^ 1 \）意义上的欧拉方程。此外，\（L ^ 1 \）还研究了这些物理量根据\（\ nu \）的收敛速度。该\（L ^ 1 \）收敛和收敛速度是由两个事实证明。一种是发明一种与迭代方法相结合的先验估计，并将其迭代到黎曼不变量的高阶导数上，以便获得\（\ partial _ {x} ^ {i} \ rho _ {\ nu} \ ）（\（i = 0,1,2 \））和\（\ partial _ {x} ^ {j} v _ {\ nu} \）（\（j = 0,1,2,3 \））必需的区域。另一个是关于特征曲线的凸度，用于根据\（\ nu \）估计特征曲线之间的距离。。这些理论结果也得到数值模拟的支持。