SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1463-1488, January 2020.
In this paper, we are concerned with the system of Euler equations with time-dependent damping like $-\frac{\mu}{(1+t)^\lambda}u$ for physical parameters $\lambda\ge 0$ and $\mu>0$. It is well known that, when $0\le \lambda<1$, the time-asymptotic-degenerate damping plays the key role which makes the damped Euler system behave like time-degenerate diffusion equations, while, when $\lambda>1$, the damping effect becomes really weak and can be neglected, which makes the dynamic system essentially behave like a hyperbolic system, and the singularity of solutions like shock waves will form. However, in the critical case $\lambda=1$, when $0<\mu\le 2$, the solutions of the system will blow up, but when $\mu>2$, the system is expected to possess global solutions. Here, we are particularly interested in the asymptotic behavior of the solutions in the critical case. By a heuristical analysis (variable scaling technique), we realize that, in this critical case, the hyperbolicity and the damping effect both play crucial roles and cannot be neglected. We first artfully construct the asymptotic profile, a special linear wave equation with time-dependent damping, which is totally different from the case of $0\leq\lambda<1,\mu>0$, whose profile is a self-similar solution to the corresponding parabolic equation. Then we rigorously prove that the solutions time-asymptotically converge to the solutions of linear wave equations with critical time-dependent damping. The convergence rates shown are optimal, by comparing with the linearized equations. The proof is based on the technical time-weighted energy method, where the time-weight is dependent on the parameter $\mu$.

中文翻译：

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临界情况下具有时变阻尼的Euler方程解的渐近行为

SIAM数学分析杂志，第52卷，第2期，第1463-1488页，2020年1月。
在本文中，我们关注具有时间相关阻尼的Euler方程组，例如物理参数$ \ lambda \ ge 0 $和$-\ frac {\ mu} {（1 + t）^ \ lambda} u $和$ \ mu> 0 $。众所周知，当$ 0 \ le \ lambda <1 $时，时间渐近退化的阻尼起着关键作用，这使得阻尼的Euler系统的行为就像时间退化的扩散方程，而当$ \ lambda> 1 $时，阻尼作用实际上变得很弱，可以忽略不计，这使得动力学系统的行为本质上像一个双曲线系统，并且将形成像冲击波这样的奇异解。然而，在临界情况下，当λ0 ＝ 1 2时，系统的解将爆炸，而当λ2，系统将具有全局解。这里，我们对临界情况下解的渐近行为特别感兴趣。通过启发式分析（可变缩放技术），我们意识到，在这种关键情况下，双曲率和阻尼效应都起着至关重要的作用，并且不能忽略。我们首先巧妙地构造了渐近曲线，这是一个特殊的线性波方程，具有随时间变化的阻尼，与$ 0 \ leq \ lambda <1，\ mu> 0 $的情况完全不同，后者的轮廓是一个自相似解。相应的抛物线方程。然后，我们严格证明了，时间渐近收敛于具有临界时间相关阻尼的线性波动方程的解。通过与线性化方程比较，所示收敛速度是最佳的。证明基于技术时间加权能量法，