In this paper, we consider the compressible Euler equations with time-dependent damping $ \frac{{\alpha}}{(1+t)^\lambda}u $ in one space dimension. By constructing "decoupled" Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient grows with a algebraic rate. We study the case $ \lambda\neq1 $ and $ \lambda = 1 $ respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for $ 1<\gamma<3 $ we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.

中文翻译：

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具有瞬态阻尼的可压缩欧拉方程的奇异性形成

在本文中，我们考虑在一维空间中具有瞬态阻尼 $ \frac{{\alpha}}{(1+t)^\lambda}u $ 的可压缩欧拉方程。通过构造光滑解的“解耦”Riccati 型方程，我们提供了一些充分条件，在这些条件下经典解必须在有限时间内分解。作为副产品，我们证明了导数爆炸，有点像冲击波的形成，如果初始数据的导数在某一点适当大，即使阻尼系数以代数速率增长。我们分别研究了 $\lambda\neq1 $ 和 $\lambda = 1 $ 的情况，而且我们的结果对解的大小和初始黎曼不变量的正/单调性没有限制。此外，对于 $1<\gamma<