Navier-Stokes方程与Euler方程的稳定性比较

比较了Navier-Stokes 方程和Euler方程的稳定性;并以它们的典型初值问题为例,分析了Navier-Stokes方程和Euler方程稳定性不同的原因．     

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known twodimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 35–48, July, 2009.     

VISCOSITY METHOD OF A NON－HOMOGENEOUS BURGERS EQUATION

Abstract In [1], Ding et al. studied the nonhomogeneous Burgers equation ut uux = μuxx 4x.(1.1) This paper will prove that when μ → 0 the solution of (1.1) will approach the generalized solution of ut uux = 4x.(1.2) The authors notice that the equation (1.2) is beyond the scope of investigations by Oleinik O. in [2]. The solutions here are unbounded in general. The paper also studies the δ-wave phenomenon when (1.2) is jointed with some other equation.     

常系数线性差分方程的微分解法

用微分方法 ,解常系数线性差分方程     

REMARK ON UNIQUE CONTINUATION OF SOLUTIONS TO THE STOKES AND THE NAVIER-STOKES EQUATIONS

1IntroductionUnique continuation of solutions to the linear partial di?erential equations with analyticcoe?cients is well known.There are more general results in elliptic,parabolic and hyperbolicequations(cf.[8-10,12-13]and references therein).The continu…     

New Bounds for the Inhomogenous Burgers and the Kuramoto-Sivashinsky Equations

We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from a previous paper of the third author, at the same time slightly improving the result. That result relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of recent results obtained by Golse and Perthame.     

“Pexiderized” homogeneity almost everywhere

In the paper we examine Pexiderized ?-homogeneity equation almost everywhere. Assume that G and H are groups with zero, (X,G) and (Y,H) are a G- and an H-space, respectively. We prove, under some assumption on (Y,H), that if functions and satisfy Pexiderized ?-homogeneity equation

F₁(αx)=?(α)F₂(x)     

Validity of the KdV equation for the modulation of periodic traveling waves in the NLS equation

Bethuel et al.  and  and Chiron and Rousset [3] gave very nice proofs of the fact that slow modulations in time and space of periodic wave trains of the NLS equation can approximately be described via solutions of the KdV equation associated with the wave train. Here we give a much shorter proof of a slightly weaker result avoiding the very detailed and fine analysis of ,  and . Our error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like method, and energy estimates.     

Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

New oscillation and nonoscillation criteria are established for the equation

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

Solutions of the three-dimensional sine-Gordon equation

We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α) and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These are an algebraic system of equations in the case of one ansatz. In the case of two ansatzes, the system of algebraic equations is supplemented by first-order ordinary differential equations. The resulting solutions U(x, y, z, t) have an important property, namely, the superposition principle holds for the function tan(U/4). The suggested approach can be used to solve the generalized sine-Gordon equation, which, in contrast to the classical equation, additionally involves first-order partial derivatives with respect to the variables x, y, z, and t, and also to integrate the sinh-Gordon equation. This approach admits a natural generalization to the case of integration of the abovementioned types of equations in a space with any number of dimensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 370–377, March, 2009.     

On reflection principles supported on a finite set

We investigate the existence of reflection formulas supported on a finite set. It is found that for solutions of the Laplace and Helmholtz equation there are no finitely supported reflection principles unless the support is a single point. This confirms that in order to construct a reflection formula that is not ‘point to point’, it is necessary to consider a continuous support. For solutions of the wave equation ^∂2u/∂x∂y=0, there exist finitely supported reflection principles that can be constructed explicitly. For solutions of the telegraph equation ^∂2u/∂x∂y+λ²u=0, we show that if a reflection principle is supported on less than five points then it is a point to point reflection principle.     

Symbolic computation of solutions for a forced Burgers equation

In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole-Hopf transformation and the traveling wave method.     

Modified method of simplest equation and its application to nonlinear PDEs

We search for traveling-wave solutions of the class of PDEswhere A_p(Q),B_r(Q),C_s(Q),D_u(Q) and F(Q) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto-Sivashinsky equation, reaction-diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.     

Approximate Solution of the Kuramoto-Shivashinsky Equation on an Unbounded Domain

The main goal of this paper is to approximate the Kuramoto-Shivashinsky (K-S for short) equation on an unbounded domain near a change of bifurcation, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider PDEs with quadratic nonlinearities and derive rigorously the modulation equation, which is called the Ginzburg-Landau (G-L for short) equation, for the amplitudes of the dominating modes.     

The stability of Cauchy's gamma-beta functional equation

We obtain the super stability of Cauchy's gamma-beta functional equation

B(x,y)f(x+y)=f(x)f(y),     

On the Matrix Equation X^{ - 1} X* = A

Solvability conditions are examined for the matrix equation , which cannot be found in the well-known reference books on matrix theory. Methods for constructing solutions to this equation are indicated.     

Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z ₀ be a subset of Z such than n∈Z ₀ implies n + 1 ∈Z ₀. Denote the space of all compactly supported distributions by D′, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G _n and H _n , n∈Z ₀, in D′, define the corresponding nonstationary nonhomogeneous refinement equation Φ _n =H _n *Φ _n+1 (A·)+G _n for all n∈Z ₀ where Φ _n , n∈Z ₀, is in a bounded set of D′. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ _n , n∈Z ₀, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations for all n∈Z ₀ where the matrices S _n and the vectors , n∈Z ₀, can be constructed explicitly from H _n and G _n respectively. The results above are still new even for stationary nonhomogeneous refinement equations. Received December 30, 1999, Accepted June 15, 2000