Multi-d isothermal Euler flow: Existence of unbounded radial similarity solutions
Multi-d isothermal Euler flow: Existence of unbounded radial similarity solutions
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DOI：
10.1016/j.physd.2020.132511
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发表时间：
2020-09-01
影响因子：
4
通讯作者：
Tsikkou, Charis
中科院分区：
文献类型：
--
作者：
Jenssen, Helge Kristian;Tsikkou, Charis
我们证明了理想多向气体等温流动的多维可压缩欧拉系统承认由于波聚焦而具有无界振幅的全局实时径向对称解。这些例子都是相似解，并且涉及一个聚焦于原点的收敛波。在坍缩时，密度而不是速度变得无界，导致激波膨胀。这些解被构造为到原点r的径向距离和时间t的函数。我们验证它们为原始的、多维的、等温的欧拉系统提供了真正的弱解。虽然在理想气体的全欧拉系统的著名的古德利解的激励下，我们考虑的解是不同类型的。在古德利解中，入射激波通过在零压力下穿透静止的“冷”气体向原点传播（由于激波上游的温度消失，没有反压力），在坍缩时伴随着速度和压力的增加，而不是密度的增加。目前尚不清楚在没有零压力区域的情况下，整个系统是否允许无界解。目前的工作表明，简化的等温模型确实承认这种行为。(C) 2020 Elsevier B.V.版权所有
We show that the multi-dimensional compressible Euler system for isothermal flow of an ideal, polytropic gas admits global-in-time, radially symmetric solutions with unbounded amplitudes due to wave focusing. The examples are similarity solutions and involve a converging wave focusing at the origin. At time of collapse, the density, but not the velocity, becomes unbounded, resulting in an expanding shock wave. The solutions are constructed as functions of radial distance to the origin r and time t. We verify that they provide genuine, weak solutions to the original, multi-d, isothermal Euler system.While motivated by the well-known Guderley solutions to the full Euler system for an ideal gas, the solutions we consider are of a different type. In Guderley solutions an incoming shock propagates toward the origin by penetrating a stationary and "cold" gas at zero pressure (there is no counter pressure due to vanishing temperature upstream of the shock), accompanied by blowup of velocity and pressure, but not of density, at collapse. It is currently not known whether the full system admits unbounded solutions in the absence of zero-pressure regions. The present work shows that the simplified isothermal model does admit such behavior. (C) 2020 Elsevier B.V. All rights reserved.