On the operator of exterior derivation on the Riemannian manifolds with cylindrical ends

We formulate some conditions for the normal and compact solvability of the operator of exterior derivation on the cylindrical manifolds equipped with some Riemannian metrics. Some analogous results were obtained in the particular case of warped cylinders [1].     

On existence and nonexistence of global solutions of Cauchy–Goursat problem for nonlinear wave equations

We consider the Cauchy–Goursat initial characteristic problem for nonlinear wave equations with power nonlinearity. Depending on the power of nonlinearity and the parameter in an equation we investigate the problem on existence and nonexistence of global solutions of the Cauchy–Goursat problem. The question on local solvability of the problem is also considered.     

三维Leibniz代数的分类

Leibniz代数是比Lie代数更广泛的一类代数,它通常不满足反交换性.在这篇文章里我们确定了维数等于3的Leibniz代数的同构类.     

Some remarks on solvability and various indices for implicit differential equations

It has been shown [17,18,21] that the notion of index for DAEs (Differential Algebraic Equations), or more generally implicit differential equations, could be interpreted in the framework of the formal theory of PDEs. Such an approach has at least two decisive advantages: on the one hand, its definition is not restricted to a “state-space” formulation (order one systems), so that it may be computed on “natural” model equations coming from physics (which can be, for example, second or fourth order in mechanics, second order in electricity, etc.) and there is no need to destroy this natural way through a first order rewriting. On the other hand, this formal framework allows for a straightforward generalization of the index to the case of PDEs (either “ordinary” or “algebraic”). In the present work, we analyze several notions of index that appeared in the literature and give a simple interpretation of each of them in the same general framework and exhibit the links they have with each other, from the formal point of view. Namely, we shall revisit the notions of differential, perturbation, local, global indices and try to give some clarification on the solvability of DAEs, with examples on time-varying implicit linear DAEs. No algorithmic results will be given here (see [34,35] for computational issues) but it has to be said that the complexity of computing the index, whatever approach is taken, is that of differential elimination, which makes it a difficult problem. We show that in fact one essential concept for our approach is that of formal integrability for usual DAEs and that of involution for PDEs. We concentrate here on the first, for the sake of simplicity. Last, because of the huge amount of work on DAEs in the past two decades, we shall mainly mention the most recent results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Strong solvability of 3-D Cahn–Hilliard system in elastic solids

In this paper we prove the existence and uniqueness of a strong solution to a 3-D model of phase separation in elastic solids. The model has the form of an initial-boundary-value problem for a nonlinear coupled system of hyperbolic–parabolic type. The key idea of the proof is based on the analysis of the system once- and twice differentiated with respect to time variable. The paper develops results of the previous work (Top. Meth. Nonlin. Anal., in press). Copyright © 2007 John Wiley & Sons, Ltd.     

Exact and approximate resolution of integral multiflow and multicut problems: algorithms and complexity

This is a summary of the author’s PhD thesis supervised by Marie- Christine Costa and Frédéric Roupin and defended on 20 November 2006 at the Conservatoire National des Arts et Métiers in Paris (France). The thesis is written in French and is available upon request from the author. This work deals with two well-known optimization problems from graph theory: the maximum integral multiflow and the minimum multicut problems. The main contributions of this thesis concern the polynomial-time solvability and the approximation of these two problems (and of some of their variants) in classical classes of graphs: bounded tree-width graphs, planar graphs and grids, digraphs, etc.      

Theoretical analysis of a model of the thermal boundary layer with drag

For solutions of a system of degenerate quasilinear parabolic equations we prove some interior Schauder estimates and use them to establish an existence theorem for solutions of the problem of extending the thermal boundary layer of a compressible fluid in a magnetic field.     

Local solvability of the cauchy problem of a fifth-order nonlinear dispersive equation

The solvability of the fifth-order nonlinear dispersive equation δtu＋au （δxu）^2＋βδx^3u＋γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5（R） and s〉1/4,then the Cauchy problem has a unique solution in C（[-T,T],H^5（R）） for some T〉0.     

On Counting the Number of Eigenvalues in the Right Half-Plane for Spectral Problems Connected with Hyperbolic Systems. II. Differential Equations

This article is an immediate continuation of [1]. Solution of the Lyapunov equation leads to a boundary value problem for the first-order hyperbolic equations in two variables with data on the boundary of the unit square. In general, the problems of this kind are not normally solvable. We prove that the boundary value problems in question possess the Fredholm property under some conditions.     

A limit-point criterion for a class of Sturm-Liouville operators defined in spaces

Using a recent result of Chernyavskaya and Shuster we show that the maximal operator determined by on , -\infty$">, where and the mean value of computed over all subintervals of of a fixed length is bounded away from zero, shares several standard ``limit-point at " properties of the case. We also show that there is a unique solution of that is in all , .