Related papers: On the behavior of multidimensional axisymmetric s…
We study the affine solutions of the equations of plane oscillations of cold plasma, which, under the assumption of electrostaticity, correspond to the Euler-Poisson equations in the repulsive case. It is proved that the zero equilibrium…
We study the influence of the friction term on the radially symmetric solutions of the repulsive Euler-Poisson equations with a non-zero background, corresponding to cold plasma oscillations in many spatial dimensions. It is shown that for…
We prove that arbitrary smooth perturbations of the zero equilibrium state of the repulsive pressureless Euler-Poisson equations, which describe the behavior of cold plasma, blow up for any non-constant doping profile already in…
In this paper, the finite time blow-up of smooth solutions to the Cauchy problem for full Euler-Poisson equations and isentropic Euler-Poisson equations with repulsive forces or attractive forces in high dimensions $(n\geq3)$ is proved for…
The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two…
In this paper, we continue to study the blowup problem of the $N$-dimensional compressible Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. In details, we extend the recent result of "M.W. Yuen, \textit{Blowup for…
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…
It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a…
In this paper, we study the blowup of the $N$-dim Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. We provide a novel integration method to show that the non-trivial classical solutions $(\rho,V)$, with compact…
The motion of a compressible inviscid radiative flow can be described by the radiative Euler equations, which consists of the Euler system coupled with a Poisson equation for the radiative heat flux through the energy equation. Although…
We prove the finite time blow-up for $C^1$ solutions to the Euler-Poisson equations in $\Bbb R^n$, $n\geq 1$, with/without background density for initial data satisfying suitable conditions. We also find a sufficient condition for the…
The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure $p$, the spatial part $\underline{u} \in \R^3$ of the dimensionless four-velocity and the particle density $n$. Radially symmetric solutions of…
In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.
This article is the continued version of the analytical blowup solutions for 2-dimensional Euler-Poisson equations \cite{Y1}. With extension of the blowup solutions with radial symmetry for the isothermal Euler-Poisson equations in $R^{2}$,…
We prove finite-time vorticity blowup for smooth solutions of the 2D compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by…
We study the formation of singularity for the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic…
In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of…
In this paper, for compressible Euler equations in multiple space dimensions, we prove the break-down of classical solutions with a large class of initial data by tracking the propagation of radially symmetric expanding wave including…
We consider the Euler-Poincar\'e equation on $\mathbb R^d$, $d\ge 2$. For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu \cite{Chae…
We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form)…