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Singularity formation for compressible Euler equations with time-dependent damping

This work is supported in part by the National Natural Science Foundation of China (Grant No. 11671237)

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  • 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.

    Mathematics Subject Classification: Primary: 35Q31, 76N10; Secondary: 76L05.

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