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

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}% 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 $f\ge 0$ and $f\in C^1;$ $s\ge 1$, $a_1>0$ 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.

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

References in corpus (8)

• Analytical Solutions to the Navier-Stokes Equations
• Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars
• Analytical Solutions to the Navier-Stokes Equations with Density-dependent Viscosity and with Pressure
• Analytical Blowup Solutions to the Pressureless Navier-Stokes-Poisson Equations with Density-dependent Viscosity in R^N
• Analytically Periodic Solutions to the 3-dimensional Euler-Poisson Equations of Gaseous Stars with Negative Cosmological Constant
• Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations
• Self-Similar Blowup Solutions to the 2-Component Degasperis-Procesi Shallow Water System
• Analytical Blowup Solutions to the Isothermal Euler-Poisson Equations of Gaseous Stars in R^N

Cited by in corpus (6)

• Self-Similar Solutions with Elliptic Symmetry for the Compressible Euler and Navier-Stokes Equations in R^{N}
• Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces
• Exact, Rotational, Infinite Energy, Blowup Solutions to the 3-Dimensional Euler Equations
• Self-Similar Blowup Solutions to the 2-Component Degasperis-Procesi Shallow Water System
• Perturbational Blowup Solutions to the 2-Component Camassa-Holm Equations
• The Inviscid, Compressible and Rotational, 2D Isotropic Burgers and Pressureless Euler-Coriolis Fluids; Solvable models with illustrations