Observation of Generalized Euler Equations

We investigate conditons for generalized double bracket flows to be Morse–Smale.The results yield necessary and sufficient conditions for observability in finite time. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unstable Manifolds of Euler Equations

We consider a steady state v₀ of the Euler equation in a fixed bounded domain in ?ⁿ. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center‐stable subspaces. By rewriting the Euler equation as an ODE on an infinite‐dimensional manifold of volume‐preserving maps in W^{k, q} the unstable (and stable) manifolds of v₀ are constructed under a certain spectral gap condition that is satisfied for both two‐dimensional and three‐dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v₀ in the sense that arbitrarily small W^{k, q} perturbations can lead to L² growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.     

Homogenization of Incompressible Euler Equations

In this paper, we perform a nonlinear multiscale analysis for incompressible Euler equations with rapidly oscillating initial data. The initial condition for velocity field is assumed to have two scales. The fast scale velocity component is periodic and is of order one.One of the important questions is how the two-scale velocity structure propagates in time and whether nonlinear interaction will generate more scales dynamically. By using a Lagrangian framework to describe the propagation of small scale solution, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homogenized equation for the incompressible Euler equations. Preliminary numerical experiments are presented to demonstrate that the homogenized equation captures the correct averaged solution of the incompressible Euler equation.     

Generalized Euler and Chu-Vandermonde identities

The object of this paper is to present two surprisingly general identities involving the generating functions of the number of inversions between multisets (of not necessarily uniform multiplicities). The first identity is a generalization of the Chu-Vandermonde identity, and the second a well-known identity of Euler. Our proofs are based on a one-to-one correspondence between multisubsets and certain permissible paths in a digraph with monomial weights.     

Generalized Euler equations in quaternions

The quaternion analog of the Euler dynamic equations obtained in [1–3] is generalized to the case of an arbitrary trihedron connected with a body.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1414–1416, October, 1994.     

Property (X) for Second-Order Nonlinear Differential Equations of Generalized Euler Type

In this paper the generalized nonlinear Euler differential equation t²k(tu′)u″ + t(f(u)+ k(tu′))u′ + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. We present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property (X⁺), which is very important for the existence of periodic solutions and oscillation theory.     

On Some Model Equations of Euler and Navier-Stokes Equations

The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line(vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between tur...     

Euler Characteristics for Generalized Tetrahedron Groups

This paper is concerned with a class of groups which generalize the ordinary tetrahedron groups. This class of groups was introduced by Fine and Rosenberger and independently by Vinberg. Some conditions for the generalized tetrahedron groups to have a rational Euler characteristic are shown here and the values of the Euler characteristics in these cases are calculated. In addition here are some geometrical applications of the Euler characteristic of generalized tetrahedron groups.^{1 2}     

Generalized Neutrino Equations

I discuss generalized spin-1/2 equations for neutrinos. They have been obtained by means of the Gersten’s method for derivation of arbitrary-spin relativistic equations. Possible physical consequences are discussed.     

On the Ultra Relativistic Euler Equations

We consider the relativistic Euler equations in isentropic fluids with the equation of state , which is the ultra-relativistic limit. We analyze the single shocks. We study the shock interaction, and give explicit example for the non-backward uniqueness. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

有关二维Euler方程的一些估计

首先得到Ｌｏｒｅｎｔｚ空间中的一些结果,然后在此基础上得到了有关二维Ｅｕｌｅｒ方程解的一些估计。这些估计与该方程当初始旋度ω０∈Ｌ＾－１∩Ｌ＾ｐ（ｐ〉１）时解的唯一性有关。     

求解Euler方程的高精度熵相容格式

熵相容格式相比于一般的熵稳定格式进一步控制了激波处的熵增量,一维情况下能有效消除膨胀激波及间断处的伪振荡等现象.对于Euler方程,可以通过对特征变量进行WENO重构以获得高阶熵相容格式的数值粘性项,然后与高阶熵守恒格式结合得到高精度熵相容格式,在WENO重构过程中的权重关于特征变量是非线性的,这导致了大量的向量内积运算.通过用压强和熵代替特征变量来计算权重,可以显著减少重构的计算量,并且数值算例表明这种权重的计算方式能很好地保持数值格式的高阶精度和基本无振荡的效果.     

Inverse Cascade Solutions of the Euler Equations

The motion of an ideal incompressible fluid in a two-dimensional domain M is considered. We assume that the initial velocity field is small-scaled, i.e., its Fourier transform is concentrated at high frequences. The extreme case of flows corresponding to solutions of the Euler equations starting from the zero scale is studied. The main result of the paper is that such a solution exists. Its construction uses the variational principle, generalized flows, and continual braids. Bibliography: 5 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 238–244.     

三类包含Euler函数的方程

讨论了三类包含Euler函数的方程x-ψ(x)=2~(ω(x)),x-ψ(ψ(x))=2~(ω(x))与ψ(x~k)=2~(ω(x~k))的可解性,利用初等方法给出这三类方程的所有正整数解,其中ψ(x)为Euler函数,ω(x)为x的相异素因子个数.     

Exterior Problem for the Three-dimensional Euler Equations

A priori estimates for the exterior initial boundary value problems of the Euler equations are considered. The existence and uniqueness of a local solution is proved.     

广义m阶Bernoulli数和广义m阶Euler数的计算公式

使用发生函数方法,利用第一类Stirling数和第二类Stirling数分别给出广义m阶Bernoulli数和广义m阶Euler数的计算公式.     

Entropies and Flux-Splittings for the Isentropic Euler Equations

51. IntroductionThe Euler equations for an iselitropic compressible fluid readwhere p 2 0 denotes the density, v the velocity) and p(p) 2 0 the pressure. The equstiope(1.1) form a nonlinear hyperbolic system of conserVation laws. By definition, a msthematicalentropy n = n(p, v) and its corresponding elltropy flux-function q = q(p, v) satisfyfor any smooth solution (p,m) of (1.1). A weak entropy3 by definition, vanishes on thevacuum p = 0. Following Laxlll'lz], we are interested in measuxable…     

一维可压缩Euler方程组的两个模型

作者考察了一维可压缩Euler方程组的两个模型.利用特征分解和Gronwall不等式,首先得到具有几何结构且绝热指数γ=3的一维可压缩Euler方程组L~∞模的一致有界性.进一步,考虑当绝热指数γ=-1时,一维非等熵可压缩Euler方程组的Cauchy问题.在适当的假设下,得到该系统的整体经典解.     

On the Natural Growth Quasilinear Elliptic Euler Equations

In this paper, we consider the eigenvalue problem and the Dirichlet problem of general Euler equations under the natural growth condition.