We first study the global-in-time existence of strong solutions to a one-dimensional system modeling the movement of a piston in a viscous compressible gas. Moreover, we prove the asymptotic stability of the solution toward a chosen constant state (in the sense that we can impose the final position of the piston, the final densities being fixed by the conservation of mass and the choice of the final position) thanks to a constant force acting in the equation of the point mass whose expression depends explicitly of the chosen final position. The norm of the solution in the function space of the initial data decays exponentially toward this constant state. Then, we prove the existence of weak solutions to this system for initial velocity in the energy state and for the initial density with bounded total variation. The weak solution is unique and also decay exponentially toward the chosen constant state thanks to the same constant force acting on the point mass. We use the result of existence of strong solutions to prove the existence of weak solutions, whereas the result on exponential decay of weak solution is independent of the one for the strong solutions.

中文翻译：

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建模粘性流体中活塞运动的系统的弱解

我们首先研究一维系统的强解的全局时间存在性，该一维系统模拟了活塞在粘性可压缩气体中的运动。此外，由于以下原因，我们证明了溶液在选定的恒定状态下的渐近稳定性（从某种意义上说，我们可以施加活塞的最终位置，而最终密度由质量守恒和最终位置的选择来固定）。在点质量方程中作用的恒力，其表达式明确取决于所选的最终位置。初始数据的函数空间中解的范数朝此恒定状态呈指数衰减。然后，我们证明了该系统对于能量状态下的初始速度和具有有限总变化的初始密度的弱解的存在。弱解是唯一的，并且由于作用在点质量上的恒定力，它也朝选定的恒定状态呈指数衰减。我们使用强解的存在结果来证明弱解的存在，而弱解的指数衰减结果与强解的存在无关。