论文标题
新的自相似欧拉流：梯度灾难没有冲击形成
New Self-similar Euler Flows: gradient catastrophe without shock formation
论文作者
论文摘要
我们考虑了在两个和三个空间尺寸中理想气体的完整压缩欧拉系统的自相似解决方案。该系统承认，根据参数$λ$和$κ$，有一个相似解决方案的2参数家族。 需要本地有限的质量，动量和能量意味着对$λ$和$κ$的某些限制。针对特定类型的流程施加了进一步的限制。例如，古德利（Guderley）的开创性构造无界的融合冲击波入侵了静态流体，需要$κ= 0 $和$λ> 1 $。 在这项工作中，我们分析了制度$ 0 <λ<1 $，这似乎以前没有解决。我们的发现包括：（i）Guderley Shock Solutions的不存在； （ii）在3-D中存在有界和连续的相似性流，提供$κ$的值$ \hatκ= \ frac {2（1-λ）} {γ-1} $，$λ$，$λ$足够小，$γ$足够大; （iii）后者的延续是由于全球定义和连续的相似性解决方案而超出了崩溃的流动。 这些解决方案的一个关键特征是，与古德利解决方案相比，它们在崩溃时保持界限，而密度，速度和声速都遭受了梯度爆炸。值得注意的是，尽管崩溃时无限梯度，但没有出现冲击波。该分析基于分析和数值计算的组合。
We consider self-similar solutions to the full compressible Euler system for an ideal gas in two and three space dimensions. The system admits a 2-parameter family of similarity solutions depending on parameters $λ$ and $κ$. Requiring locally finite amounts of mass, momentum, and energy imply certain constraints on $λ$ and $κ$. Further constraints are imposed for particular types of flows. E.g., Guderley's pioneering construction of an unbounded converging shock wave invading a quiescent fluid, requires $κ=0$ and $λ>1$. In this work we analyze the regime $0<λ<1$, which does not appear to have been addressed previously. Our findings include: (i) non-existence of Guderley shock solutions; (ii) existence of bounded and continuous incoming similarity flows in 3-d provided $κ$ takes the value $\hatκ=\frac{2(1-λ)}{γ-1}$, $λ$ is sufficiently small, and $γ$ is sufficiently large; (iii) continuation of the latter flows beyond collapse as globally defined and continuous similarity solutions. A key feature of these solutions is that they, in contrast to Guderley solutions, remain bounded at time of collapse, while the density, velocity, and sound speed all suffer gradient blowup. It is noteworthy that, notwithstanding infinite gradients at collapse, no shock wave appears. The analysis is based on a combination of analytical and numerical calculations.
