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.

我们表明，理想的多方气体的等温流的多维可压缩Euler系统允许由于波聚焦而具有全局无限制幅度的全局时间径向对称解。这些示例是相似性解决方案，涉及聚焦于原点的会聚波。在坍塌时，密度而不是速度变得不受限制，从而导致膨胀的冲击波。这些解被构造为到原点的径向距离的函数$\mathrm{[R}$ 和时间 $Ť$。我们验证了它们为原始的多维等温欧拉系统提供了真正的弱解决方案。尽管受到众所周知的Guderley解决方案对理想气体的整个Euler系统的激励，但我们认为的解决方案却是另一种类型。在Guderley解决方案中，传入的冲击通过在零压力下渗透固定的“冷”气体（由于冲击上游的温度没有消失而产生反压）而传播到原点，但伴随着速度和压力的爆炸，但密度没有爆炸，在崩溃时。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允许这种行为。