Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars
| | Speaker: | 黎海彤，长春工业大学
| | Inviter: | 王益 | Title:
| Compressible Euler Equations with Time-Dependent Damping | Time & Venue:
| 2021.06.10 14:00-16:00 思源楼809教室 | Abstract:
| In this talk, we consider the Cauchy problem for the 1D compressible Euler equations with time-dependent damping trem $ -\frac{\mu}{(1+t)^\lambda}u $. Firstly, we consider the large time behavior of solutions for $ \mu=1,-1\leq\lambda<1 $. We show that the system has a couple of global solutions uniquely, and such solutions tend time-asymptotically to the shifted nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation governed by the Darcy's law. We further derive the optimal convergence rates when the initial perturbations are in $ L^2 $.
Secondly, we consider the global and blow-up solutions for $ \lambda>0 $. For $ 0<\lambda <1 $ and $ \mu>0 $, or $ \lambda =1 $ but $ \mu>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. 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. For $ \lambda >1 $ and $ \mu>0 $, or $ \lambda =1 $ but $ 0<\mu\leq 1 $, the derivatives of solutions will blow up for all initial data.
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