Abstract
In this paper, the vacuum and singularity formation problem are considered for the compressible Euler equations with general pressure law and time-dependent damping. Firstly, the lower bound estimates of density for arbitrary classical solutions is shown for some kind of pressure functions. Then, three sufficient conditions, under which the classical solutions must break down in finite time, are shown by delicate analysis of decoupled Riccati type equations. The assumptions on pressure function automatically satisfied for gas dynamics with γ-law. Furthermore, our results have no limits on the size of the solutions or the positive/monotonicity on the initial Riemann invariants.
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