TY - JOUR

T1 - Multi-d isothermal Euler flow

T2 - Existence of unbounded radial similarity solutions

AU - Jenssen, Helge Kristian

AU - Tsikkou, Charis

N1 - Funding Information:
This work was supported in part by the National Science Foundation awards DMS-1813283 (Jenssen) and DMS-1714912 (Tsikkou). The authors would like to thank an anonymous referee whose comments substantially improved the presentation.

PY - 2020/9

Y1 - 2020/9

N2 - 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.

AB - 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.

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U2 - 10.1016/j.physd.2020.132511

DO - 10.1016/j.physd.2020.132511

M3 - Article

AN - SCOPUS:85083524966

VL - 410

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

M1 - 132511

ER -