论文标题
与时间相关阻尼的3D可压缩欧拉方程的移位冲击形成
Formation of shifted shock for the 3D compressible Euler equations with time-dependent damping
论文作者
论文摘要
在本文中，我们在三个空间维度中，在没有任何对称条件的情况下，在三个没有任何对称条件下，在三个空间尺寸中，我们将冲击形成显示为可压缩的Euler方程$ \ frac {a \ p u} {（1+t）^{\ lam}} $。众所周知，对于$ \ lam> 1 $，阻尼太弱，无法防止适当大数据的冲击形成。但是，经典结果仅显示了溶液的有限存在。遵循D.Christodoulou在\ cite {Christodoulou2007}中的工作，从最初的等式和无关的短脉冲数据开始，我们表明，震动的形成的特征是特征性高空的崩溃，并且消失了反向叶面密度的功能$μ$，并在其最初的derivation and derivent and dostel and lif and lif and distion and and distion and distion and distion and distion and distion and distion。 $ t _ {\ ast}（a，\ lam）$呈指数级。此外，阻尼效果将改变冲击形成的时间$ t _ {\ ast} $。本文中的方法也可以扩展到具有一般时间延迟阻尼的Euler方程。
In this paper, we show the shock formation to the compressible Euler equations with time-dependent damping $\frac{a\p u}{(1+t)^{\lam}}$ in three spatial dimensions without any symmetry conditions. It's well-known that for $\lam>1$, the damping is too weak to prevent the shock formation for suitably large data. However, the classical results only showed the finite existence of the solution. Follow the work by D.Christodoulou in\cite{christodoulou2007}, starting from the initial isentropic and irrotational short pulse data, we show the formation of shock is characterized by the collapse of the characteristic hypersurfaces and the vanishing of the inverse foliation density function $μ$, at which the first derivatives of the velocity and the density blow up, and the lifespan $T_{\ast}(a,\lam)$ is exponentially large. Moreover, the damping effect will shift the time of shock formation $T_{\ast}$. The methods in the paper can also be extended to the Euler equations with general time-decay damping.
