Boltzmann-Poisson方程组初边值问题的整体弱解

§ 1.　Introduction　　LetΩ R3beasmoothboundedopenset,f(t,x ,v)bethedistributionfunctionofcar riersattimet >0 ,positionvectorx∈Ωandvelocityvectorv∈R3.Theforcefieldactingonthecarriersisdenotedbythe3 dimensionalvectorfunctionE(t,x) ,thepotentialcorespondingtoE(t,x)isdenotedbyu(t,x) ,then (f,E)satisfythefollowingBoltzmann Poissonsystem tf +v· xf +E· vf =Q(f) ,　 (t,x ,v)∈R+ ×Ω×R3,(1 .1 )E =- xu ,-Δxu =ρ ,　 (t,x) ∈R+ ×Ω ,(1 .2 )whereρ(t,x) =∫R3 f(t,x ,v)dvisthecarriernum…     

The Cauchy Problem for the System of Maxwell Equations

We consider the problem of analytic continuation of a solution to the system of Maxwell equations in a bounded spatial domain from data on part of the boundary of the domain. We construct an approximate solution to the problem using the Carleman matrix method.     

On a Problem of Karpilovsky

Let G be a finite elementary group. Let ⁿ (G) denote the nth power of the augmentation ideal (G) of the integral group ring G. In this paper, we give an explicit basis of the quotient group Q_n(G) = ⁿ(G)/ⁿ⁺¹ (G) and compute the order of Qⁿ (G).2000 Mathematics Subject Classification: 16S34, 20C05     

On the Application of the Auxiliary Problem Principle

The auxiliary problem principle (APP) derives from a general theory on decomposition-coordination methods establishing a comprehensive framework for both one-level and two-level methods. In this paper, the results of the two-level methods of APP are specialized for an efficient application to some engineering problems.     

Two-Frequency Autowave Processes in the Complex Ginzburg–Landau Equation

We study the complex Ginzburg–Landau equation with zero Neumann boundary conditions on a finite interval and establish that this boundary problem (with suitably chosen parameters) has countably many stable two-dimensional self-similar tori. The case of periodic boundary conditions is also investigated.     

Viscoelastic boundary stabilization for a transmission problem in elasticity

We consider an anisotropic body which is constituted of twodifferent types of materials supporting a memory boundary conditionand we show that its energy decays uniformly as time goes toinfinity with the same rate as the relaxation function g, thatis, the energy decays exponentially when g decays exponentially,and polynomially when g decays polynomially.     

Steepest Descent, CG, and Iterative Regularization of Ill-Posed Problems

The state of the art iterative method for solving large linear systems is the conjugate gradient (CG) algorithm. Theoretical convergence analysis suggests that CG converges more rapidly than steepest descent. This paper argues that steepest descent may be an attractive alternative to CG when solving linear systems arising from the discretization of ill-posed problems. Specifically, it is shown that, for ill-posed problems, steepest descent has a more stable convergence behavior than CG, which may be explained by the fact that the filter factors for steepest descent behave much less erratically than those for CG. Moreover, it is shown that, with proper preconditioning, the convergence rate of steepest descent is competitive with that of CG.This revised version was published online in October 2005 with corrections to the Cover Date.     

Stability of solutions to Hilbert boundary value problem under perturbation of the boundary curve

In this paper, the authors discuss the stability of the solutions to Hilbert boundary value problem under perturbation of the unit circle. When the index of this problem is non-negative, by extending Lavrentjev's conformal mapping on a region approximating to a unit disc, we show the solutions are stable under small perturbations. For negative index we give a conception of quasi-solution and discuss its stability correspondingly.     

A problem of D.H. Lehmer and its mean square value formula

Let q be an odd positive integer and let a be an integer coprime to q. For each integer b coprime to q with 1?b<q, there is a unique integer c coprime to q with 1?c<q such that . Let N(a,q) denote the number of solutions of the congruence equation with 1?b,c<q such that b,c are of opposite parity. The main purpose of this paper is to use the properties of Dedekind sums, the properties of Cochrane sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the mean square value , and give a sharp asymptotic formula.     

ON THE EQUATION $\square\phi =|\nabla \phi |^2$ IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in $n$ space dimensions $$\align \square\p &=F(\partial\p ),\\p (0,x)&=f(x),\quad \partial_t\p (0,x)=g(x), \endalign$$ where $\square =\partial_t^2-\triangle$ is the wave operator, $F$ is quadratic in $\partial\p$ with $\partial =(\partial_t,\partial_{x_1},\cdots ,\partial_{x_n})$. The minimal value of $s$ is determined such that the above Cauchy problem is locally well-posed in $H^s$. It turns out that for the general equation $s$ must satisfy $$s>\max\Big(\frac{n}{2}, \frac{n+5}{4}\Big).$$ This is due to Ponce and Sideris (when $n=3$) and Tataru (when $n\ge 5$). The purpose of this paper is to supplement with a proof in the case $n=2,4$.