Stabilities for Nonisentropic Euler-Poisson Equations

Stabilities for Nonisentropic Euler-Poisson Equations

We establish the stabilities and blowup results for the nonisentropic Euler-Poisson equations by the energy method. By analysing the second inertia, we show that the classical solutions of the system with attractive forces blow up in finite time in some special dimensions when the energy is negative. Moreover, we obtain the stabilities results for the system in the cases of attractive and repul...

Discrete Euler–Poincaré and Lie–Poisson equations

In this paper, discrete analogues of Euler–Poincaré and Lie–Poisson reduction theory are developed for systems on finite dimensional Lie groupsG with Lagrangians L : TG→ R that areG-invariant. These discrete equations provide ‘reduced’ numerical algorithms which manifestly preserve the symplectic structure. The manifold G×G is used as an approximation of TG, and a discrete Langragian L : G×G→ R...

Critical Thresholds in Euler-Poisson Equations

We present a preliminary study of a new phenomena associated with the Euler-Poisson equations — the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic, O(1) critical threshold. We investigate a class of Euler-Poisson equations, ranging from one-dimensional problems...

Critical Thresholds in Euler-Poisson Equations

On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized 2D Euler equations

In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an ‘‘unstable disk theorem’’ through a simple energy–C...