数学
欧拉方程
反向欧拉法
摄动(天文学)
欧拉公式
压缩性
数学分析
工作(物理)
能量法
功能(生物学)
欧拉法
能量(信号处理)
半隐式欧拉法
应用数学
傅里叶变换
物理
机械
量子力学
统计
进化生物学
生物
摘要
.This paper is concerned with the large time behavior of the multidimensional compressible Euler equations with time-dependent overdamping of the form \(-\frac{\mu }{(1+t)^\lambda }\rho{\boldsymbol{u}}\) in \(\mathbb R^n\), where \(n\ge 2\), \(\mu \gt 0\), and \(\lambda \in [-1,0)\). This continues our previous work dealing with the underdamping case for \(\lambda \in [0,1)\). We show the optimal decay estimates of the solutions such that for \(\lambda \in (-1,0)\) and \(n\ge 2\), \(\|\rho -1\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+\lambda }{4}n}\) and \(\|\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+\lambda }{4}n-\frac{1-\lambda }{2}}\), which indicates that a stronger damping gives rise to solutions decaying optimally slower. For the critical case of \(\lambda=-1\), we prove the optimal logarithmical decay of the perturbation of density for the damped Euler equations such that \(\|\rho -1\|_{L^2(\mathbb R^n)}\approx |\ln (e+t)|^{-\frac{n}{4}}\) and \(\|\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-1}\cdot |\ln (e+t)|^{-\frac{n}{4}-\frac{1}{2}}\) for \(n\ge 7\). The overdamping 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.KeywordsEuler equationtime-dependent dampingoptimal decay ratesoverdampingMSC codes35Q3176N1035B40
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