Multi-d isothermal Euler flow: Existence of unbounded radial similarity solutions

PHYSICA D-NONLINEAR PHENOMENA

卷:410

Article

Jenssen, Helge Kristian¹  Tsikkou, Charis² 

[1] Penn State Univ, Dept Math, State Coll, PA 16802 USA

[2] West Virginia Univ, Dept Math, Morgantown, WV 26506 USA

关键词: Compressible;    Inviscid;    Unbounded solution;    Radial flow;    Similarity solution;   

DOI  :  10.1016/j.physd.2020.132511

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

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