论文标题
具有物理真空和时间依赖性阻尼的可压缩欧拉方程的全局存在和收敛到修改的Barenblatt解决方案
Global existence and convergence to the modified Barenblatt solution for the compressible Euler equations with physical vacuum and time-dependent damping
论文作者
论文摘要
在本文中，考虑了具有时间依赖性阻尼的一维压缩欧拉方程的物理真空问题的平滑解决方案。在真空边界附近，声速为$ c^{1/2} $ - 霍尔德连续。阻尼的系数取决于时间，由此表格$ \fracμ{（1+t）^λ} $，$λ$，$μ> 0 $，$-λ$在时间上衰减。假设$ 0 <λ<1 $，$ 0 <μ$或$λ= 1 $，$ 2 <μ$，我们将证明全球存在平稳的解决方案和融合到相关的多孔媒体方程式与时间相关的耗散质量和相同总质量的相关多孔媒体方程的解决方案时，当euler equertation的初始数据是一个小的解决方案时，小bar是一个小的bartertive of pertult of the smill of the small of thematt of the smill of thematt of pertult of pert the small of the small of the。还给出了密度，速度和物理真空边界的扩大速率的点收敛速率。该证明是基于拉格朗日坐标中时空加权能量估计，椭圆估计和强烈的不平等。我们的结果是在Luo-Zeng [Comm。纯应用。数学。 69（2016），没有。 7，1354-1396]，作者考虑了具有恒定抑制作用的可压缩欧拉方程的物理真空无边界问题。
In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is $C^{1/2}$-Hölder continuous. The coefficient of the damping depends on time, given by this form $\fracμ{(1+t)^λ}$, $λ$, $μ>0$, which decays by order $-λ$ in time. Under the assumption that $0<λ<1$, $0<μ$ or $λ=1$, $2<μ$, we will prove the global existence of smooth solutions and convergence to the modified Barenblatt solution of the related porous media equation with time-dependent dissipation and the same total mass when the initial data of the Euler equations is a small perturbation of that of the Barenblatt solution. The pointwise convergence rates of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on space-time weighted energy estimates, elliptic estimates and Hardy inequality in the Lagrangian coordinates. Our result is an extension of that in Luo-Zeng [Comm. Pure Appl. Math. 69 (2016), no. 7, 1354-1396], where the authors considered the physical vacuum free boundary problem of the compressible Euler equations with constant-coefficient damping.
