论文标题
$ \ Mathbb r^n $：（i）阻尼案例的最佳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 $\mathbb R^n$: (I) under-damping case
论文作者
论文摘要
本文与多维压缩的欧拉方程有关，并与$ - \fracμ{（1+t）^λ}ρ\ boldsymbol u $ in $ \ mathbb r^n $，$ n \ ge2 $，$ n \ ge2 $，$μ> 0 $> 0 $和$λ\ in [0,1,1）的$ - \fracμ{（1+t）^λ}ρ\ boldsymbol u $。当$λ> 0 $更大时，阻尼效果时间呈弱，较弱，这称为阻尼不足。我们显示了解决方案的最佳衰减估计值，以便$ \ | \ partial_x^α（ρ-1）\ | _ {l^2（\ Mathbb r^n）} \ lot（1+t）^{ - \ frac {1+λ} $ \ | \ partial_x^α\ boldsymbol u \ | _ {l^2（\ Mathbb r^n）}} \ oit（1+t）^{ - \ frac {1+λ} {2} {2} {2} {2} {2}（\ frac {n}阻尼效应影响欧拉系统的结构。与传统的观点不同的是，更强的阻尼通常会使溶液的衰减更快，在这里令人惊讶的是，我们认识到，以$ 0 \leλ<1 $ $ 0的弱阻尼增强了解决方案的更快衰减。采用的方法是技术傅立叶分析和绿色功能方法。时间依赖性阻尼引起的主要困难在于双重的：线性化操作员的傅立叶变换的非交换性排除了基本溶液的明确表达；与时间相关的进化意味着绿色矩阵$ g（t，s）$不是翻译不变的，即$ g（t，s）\ ne g（t-s，0）$。我们制定了绿色矩阵$ g（t，s）$相对于$ t $和$ s $的确切衰减行为，线性波方程和线性双曲线系统，最后得出了非线性Euler系统的最佳衰减率。
This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent damping of the form $-\fracμ{(1+t)^λ}ρ\boldsymbol u$ in $\mathbb R^n$, where $n\ge2$, $μ>0$, and $λ\in[0,1)$. When $λ>0$ is bigger, the damping effect time-asymptotically gets weaker, which is called under-damping. We show the optimal decay estimates of the solutions such that $\|\partial_x^α(ρ-1)\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+λ}{2}(\frac{n}{2}+|α|)}$, and $\|\partial_x^α\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+λ}{2}(\frac{n}{2}+|α|)-\frac{1-λ}{2}}$, and see how the under-damping effect influences the structure of the Euler system. Different from the traditional view that the stronger damping usually makes the solutions decaying faster, here surprisingly we recognize that the weaker damping with $0\leλ<1$ enhances the faster decay for the solutions. The adopted approach is the technical Fourier analysis and the Green function method. The main difficulties caused by the time-dependent damping lie in twofold: non-commutativity of the Fourier transform of the linearized operator precludes explicit expression of the fundamental solution; time-dependent evolution implies that the Green matrix $G(t,s)$ is not translation invariant, i.e., $G(t,s)\ne G(t-s,0)$. We formulate the exact decay behavior of the Green matrices $G(t,s)$ with respect to $t$ and $s$ for both linear wave equations and linear hyperbolic system, and finally derive the optimal decay rates for the nonlinear Euler system.
