SIAM Journal on Applied Mathematics, Volume 80, Issue 6, Page 2472-2495, January 2020.
We show that the compressible Euler system for isentropic gas flow admits unbounded solutions. The examples are radial flows of similarity type and describe a spherically symmetric and continuous wave moving toward the origin. At time of focusing, both the density and the velocity become unbounded at the origin. This is followed by an expanding shock wave which slows down as it interacts with the incoming flow. While unbounded radial Euler flows have been known since the work of Guderley [Luftfahrtforschung, 19 (1942), pp. 302--311], those are at the borderline of the regime covered by the Euler model: The upstream pressure field vanishes identically (either because of vanishing temperature or vanishing density there). In contrast, the solutions we build exhibit an everywhere strictly positive pressure field, demonstrating that the geometric effect of wave focusing is strong enough on its own to drive the primary flow variables to infinity.

中文翻译：

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径向各向同性欧拉流中的振幅爆破

SIAM应用数学杂志，第80卷，第6期，第2472-2495页，2020年1月。
我们表明，等熵气流的可压缩Euler系统允许无界解。这些示例是相似类型的径向流，描述了朝原点移动的球对称连续波。在聚焦时，密度和速度在原点都变得不受限制。随后是不断扩大的冲击波，当它与输入流相互作用时，它会减慢速度。自Guderley（Luftfahrtforschung，19（1942），pp.302--311）的工作以来，无界的径向欧拉流已为人所知，但这些流处于欧拉模型所涵盖的范围的边界：上游压力场完全消失（要么是因为温度消失，要么是那里的密度消失了）。相比之下，我们构建的解决方案在每个地方都表现出严格的正压力场，