Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N

Details Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N

2009-10-07

arxiv.org/abs/0910.1272v1

Commun.Nonlinear Sci.Numer.Simul.16:2993-2998,2011

Astronomy and Astrophysics

Astrophysics

Solar and Stellar Astrophysics

7 pages Key Words: Pressureless Gas, Euler Equations, Exact Solutions, Non-Radial Symmetry, Navier-Stokes Equations, Blowup, F

Scientific paper

10.1016/j.cnsns.2010.10.021

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}% \rho(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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