Global and blow-up solutions for compressible Euler equations with time-dependent damping

This paper deals with the Cauchy problem for the compressible Euler equations with time-dependent damping, where the time-vanishing damping in the form of μ(1+t)λ makes some fantastic variety of the dynamic system. For 0<λ<1 and μ>0, or λ=1 but μ>2, the solutions are proved to exist globally in time, when the derivatives of the initial data are small, but the initial data themselves can be arbitrarily large. This is the so-called challenging case of global solutions with large initial data; while, when the initial Riemann invariants are monotonic and their derivatives with absolute value are large at least at one point, then the solutions are still bounded, but their derivatives will blow up at finite time, somewhat like the singularity formed by shock waves. For λ>1 and μ>0, or λ=1 but 0<μ≤1, the derivatives of solutions will blow up even for all initial data, including the interesting case of blow-up solutions with small initial data. Here the initial Riemann invariants are monotonic. In fact, such a blow-up phenomenon is determined by the mechanism of the dynamic system itself. In order to prove the global existence of solutions with large initial data, we introduce a new energy functional related to the Riemann invariants, which crucially enables us to build up the maximum principle for the corresponding Riemann invariants, and the uniform boundedness for the local solutions. Finally, numerical simulations in different cases are carried out, which further confirm our theoretical results.