论文标题
与时间和空间相关的扰动，一维抑制压缩的欧拉方程的全局存在和爆炸
Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation
论文作者
论文摘要
在本文中，我们考虑具有时间和空间依赖性阻尼项$ -a（t，x）v $的1D EULER方程。长期以来，众所周知，当$ a（t，x）$是正常数或$ 0 $时，该解决方案在全球范围内分别存在或分别在有限时间内炸毁。我们证明这些结果在时间和空间依赖性扰动方面是不变的。我们认为系数$ a $满足以下条件$$ | a（t，x）-μ_0| \ leq a_1（t） + a_2（x），$$其中$μ_0\ geq 0 $和$ a_1 $和$ a_2 $是具有$ t $和$ x $的集成功能。在这种情况下，当$μ_0> 0 $和$μ= 0 $时，我们显示了全局存在和带有少量初始数据的爆炸。
In this paper, we consider the 1D Euler equation with time and space dependent damping term $-a(t,x)v$. It has long been known that when $a(t,x)$ is a positive constant or $0$, the solution exists globally in time or blows up in finite time, respectively. We prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient $a$ satisfies the following condition $$ |a(t,x)- μ_0| \leq a_1(t) + a_2 (x), $$ where $μ_0 \geq 0$ and $a_1$ and $a_2$ are integrable functions with $t$ and $x$. Under this condition, we show the global existence and the blow-up with small initial data, when $μ_0 >0$ and $μ=0$ respectively.
