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

TL;DR

This paper constructs exact, non-radial solutions to the pressureless Euler equations in multiple dimensions, revealing finite-time blowup scenarios and extending solutions to pressureless Navier-Stokes equations.

Contribution

It provides explicit non-radial solutions to pressureless Euler equations in R^N, including blowup solutions, and demonstrates their applicability to pressureless Navier-Stokes equations.

Findings

Solutions blow up at finite time when a2<0.

Explicit solutions involve arbitrary functions and parameters.

Solutions are valid for pressureless Navier-Stokes equations.

Abstract

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.