7 research outputs found
Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N
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 % [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 and , and
are constants\newline In particular, for , the solutions
blow up on the finite time .
Moreover, the functions (\ref{eq234}) are also the solutions to the
pressureless Navier-Stokes equations.Comment: 7 pages Key Words: Pressureless Gas, Euler Equations, Exact
Solutions, Non-Radial Symmetry, Navier-Stokes Equations, Blowup, Free
Boundar
A variational representation of weak solutions for the pressureless Euler-Poisson equations
We derive an explicit formula for global weak solutions of the one
dimensional system of pressure-less Euler-Poisson equations. Our variational
formulation is an extension of the well-known formula for entropy solutions of
the scalar inviscid Burgers' equation: since the characteristics of the
Euler-Poisson equations are parabolas, the representation of their weak
solution takes the form of a "quadratic" version of the celebrated Lax-Oleinik
variational formula. Three cases are considered. (i) The variational formula
recovers the "sticky particle" solution in the attractive case; (ii) It
represents a repulsive solution which is different than the one obtained by the
sticky particle construction; and (iii) the result is further extended to the
multi-dimensional Euler-Poisson system with radial symmetry
Convergence to Equilibrium in Wasserstein distance for damped Euler equations with interaction forces
We develop tools to construct Lyapunov functionals on the space of
probability measures in order to investigate the convergence to global
equilibrium of a damped Euler system under the influence of external and
interaction potential forces with respect to the 2-Wasserstein distance. We
also discuss the overdamped limit to a nonlocal equation used in the modelling
of granular media with respect to the 2-Wasserstein distance, and provide
rigorous proofs for particular examples in one spatial dimension
Numerical simulations of the Euler system with congestion constraint
In this paper, we study the numerical simulations for Euler system with
maximal density constraint. This model is developed in [1, 3] with the
constraint introduced into the system by a singular pressure law, which causes
the transition of different asymptotic dynamics between different regions. To
overcome these difficulties, we adapt and implement two asymptotic preserving
(AP) schemes originally designed for low Mach number limit [2,4] to our model.
These schemes work for the different dynamics and capture the transitions well.
Several numerical tests both in one dimensional and two dimensional cases are
carried out for our schemes
Pressureless Euler alignment system with control
We study a non-local hydrodynamic system with control. First we characterize
the control dynamics as a sub-optimal approximation to the optimal control
problem constrained to the evolution of the pressureless Euler alignment
system. We then discuss the critical thresholds that leading to global
regularity or finite-time blow-up of strong solutions in one and two
dimensions. Finally we propose a finite volume scheme for numerical solutions
of the controlled system. Several numerical simulations are shown to validate
the theoretical and computational results of the paper