This paper is concerned with the asymptotic behavior of solution to the damped Euler equation far away from vacuum. First of all, the global existence together with the decay rate of the solution and its derivatives are derived by the standard energy method. The decay rates of the solution operator are further improved by Green function method. Finally, the solution of damped Euler equation is confirmed to converge its best asymptotic profile when the initial data is in a certain weighted function space, which establishes the upper and lower bounds of the decay rate to the solution of the damped Euler equation. Here, we provide a new way to check the optimal decay rate of the solutions to the damped Euler equation when the initial data is set in L¹, which also could be applied to some other diffusive evolution system.

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阻尼欧拉方程解的渐近剖面

本文关注的是远离真空的阻尼欧拉方程解的渐近行为。首先，全局存在性连同解的衰减率及其导数是通过标准能量法推导出来的。格林函数法进一步提高了求解算子的衰减率。最后，确定了阻尼欧拉方程的解在初始数据处于某个加权函数空间时收敛其最佳渐近曲线，从而建立了阻尼欧拉方程解的衰减率的上下界。在这里，我们提供了一种新方法来检查当初始数据设置为L ¹时阻尼欧拉方程的解的最佳衰减率，这也可以应用于其他一些扩散演化系统。