论文标题
$ \ Mathbb r^n $：（ii）过度阻尼案例的最佳可压缩欧拉方程的最佳衰减率
Optimal decay rates of the compressible Euler equations with time-dependent damping in $\mathbb R^n$: (II) over-damping case
论文作者
论文摘要
本文与多维可压缩的欧拉方程有关，与$ - \fracμ{（1+t）^λ}ρ\ boldsymbol u $ in $ \ mathbb r^n $，$ n \ ge2 $，$ n \ ge2 $，$ usum> 0 $> 0 $> 0 $，和$λ\ in [-1,1,0 $ in [-1,1,0）。这延续了我们以前处理$λ\ [0,1）$ $λ\的工作案件的工作。我们显示了解决方案的最佳衰减估计值，以便以$λ\ in（-1,0）$和$ n \ ge2 $，$ \ |ρ-1 u \ | _ {l^2（\ Mathbb r^n）} \ ailt（1+t）^{ - \ frac {1+λ} {4} n- \ frac {1- frac {1-λ} {2}}} $，这表明较强的阻尼会引起较强的阻尼溶液衰变的溶液衰变。对于$λ= -1 $的关键情况，我们证明了阻尼的欧拉方程的密度扰动的最佳对数衰减，因此$ \ |ρ-1 \ | _ {l^2（\ Mathbb r^n）}美元过度阻尼效应降低了溶液的衰减速率慢，这使我们在通过傅立叶分析方法和绿色功能方法获得最佳衰减速率方面有一些技术困难。在这里，我们提出了一个新想法，以克服绿色功能方法和随时间加权能量方法来克服这种困难。
This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent over-damping of the form $-\fracμ{(1+t)^λ}ρ\boldsymbol u$ in $\mathbb R^n$, where $n\ge2$, $μ>0$, and $λ\in[-1,0)$. This continues our previous work dealing with the under-damping case for $λ\in[0,1)$. We show the optimal decay estimates of the solutions such that for $λ\in(-1,0)$ and $n\ge2$, $\|ρ-1\|_{L^2(\mathbb R^n)}\approx(1+t)^{-\frac{1+λ}{4}n}$ and $\|\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+λ}{4}n-\frac{1-λ}{2}}$, which indicates that a stronger damping gives rise to solutions decaying optimally slower. For the critical case of $λ=-1$, we prove the optimal logarithmical decay of the perturbation of density for the damped Euler equations such that $\|ρ-1\|_{L^2(\mathbb R^n)}\approx |\ln(e+t)|^{-\frac{n}{4}}$ and $\|\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-1}\cdot|\ln(e+t)|^{-\frac{n}{4}-\frac{1}{2}}$ for $n\ge7$. The over-damping effect reduces the decay rates of the solutions to be slow, which causes us some technical difficulty in obtaining the optimal decay rates by the Fourier analysis method and the Green function method. Here, we propose a new idea to overcome such a difficulty by artfully combining the Green function method and the time-weighted energy method.
