R^N中无压Euler方程的某些精确爆破解
Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N
摘要：
无压欧拉方程可以作为宇宙学或等离子体物理学的简单模型。本文构造了无压Euler方程组$R^{N}:$ % [c]{c}% ρ（t，\vec{x}）= \frac{f(\frac{1}{a(t)^{s}}\underset{i=1}{\overset {N}{\sum}}x_{i}^{s})}{a(t)^{N}} \text{,} \vec{u}（t，\vec{x} %）= \frac{\overset{\cdot}{a}(t)}{a(t)} \vec{x}，a（t）= a_{1} + a_{2} t的非径向对称精确解.其中任意函数$f\geq0$和$f\in C^{1};$ $s\geq1$，$a_{1}>0$和$a_{2}$是常数$.$ \newline特别地，对于$a_{2}<0$，解在有限时间$T=-a_{1}/a_{2}$上爆破。此外，函数（\ref{eq234}）也是无压Navier-Stokes方程的解。
英文摘要：
The pressureless Euler equations can be used as simple models of cosmology or plasma physics. In this paper, we construct the exact solutions in non-radial symmetry to the pressureless Euler equations in $R^{N}:$% [c]{c}% ρ(t,\vec{x})=\frac{f(\frac{1}{a(t)^{s}}\underset{i=1}{\overset {N}{\sum}}x_{i}^{s})}{a(t)^{N}}\text{,}\vec{u}(t,\vec{x}% )=\frac{\overset{\cdot}{a}(t)}{a(t)}\vec{x}, a(t)=a_{1}+a_{2}t. \label{eq234}% where the arbitrary function $f\geq0$ and $f\in C^{1};$ $s\geq1$, $a_{1}>0$ and $a_{2}$ are constants$.$\newline In particular, for $a_{2}<0$, the solutions blow up on the finite time $T=-a_{1}/a_{2}$. Moreover, the functions (\ref{eq234}) are also the solutions to the pressureless Navier-Stokes equations.
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