Measure-Valued Solutions to Initial-Boundary Value Problems for Certain Systems of Conservation Laws: Existence and Dynamics

A framework for studying initial-boundary value problems for systems of conservation laws, in what concerns to the existence of measure-valued solutions and their asymptotic behavior, is developed here with the helpful introduction of a class of flux maps which allow a rather complete treatment of these questions including systems of practical importance as those arising in multiphase flow in porous media. The systems of this class may, in general, admit umbilic points, submanifolds where genuine nonlinearity fails, as well as elliptic regions. We prove the existence of measure-valued solutions by using the vanishing viscosity method and, also, finite difference schemes. The main result about the dynamics of the measure-valued solutions is that for certain special boundary values, given by constant states, the time-averages of these m-v solutions converge weakly to the Dirac measure concentrated at those states, for a.e. space variable. The rate of convergence of the time-averages of the expected values can be estimated by properties of the flux maps only.

     

Global Solutions to Some Quasilinear Parabolic Systems in Population Dynamics

In this present paper, using the duality technique and the Hölder's inequality, we study the global existence of the solutions to some quasilinear parabolic systems with the cross-diffusion effects in population dynamics.     

Attractors of Weakly Asymptotically Compact Set-Valued Dynamical Systems

The existence of weak attractors is established for set-valued dynamical systems which are weakly asymptotically compact and weakly dissipative. Here weak properties mean with respect to at least one trajectory for each initial value. A condition ensuring the uniqueness of such weak attractors is given. The results are illustrated with an application involving a partial differential inclusion in bounded and unbounded domains. Mathematics Subject Classifications (2000) 34D45, 37B25, 37B75, 58C06.     

A Solution Concept for Fuzzy Multiobjective Programming Problems Based on Convex Cones

A solution concept for fuzzy multiobjective programming problems based on ordering cones (convex cones) is proposed in this paper. The notions of ordering cones and partial orderings on a vector space are essentially equivalent. Therefore, the optimality notions in a real vector space can be elicited naturally by invoking a concept similar to that of the Pareto-optimal solution in vector optimization problems. We introduce a corresponding multiobjective programming problem and a weighting problem of the original fuzzy multiobjective programming problem using linear functionals so that the optimal solution of its corresponding weighting problem is also the Pareto-optimal solution of the original fuzzy multiobjective programming problem.     

可压Navier-Stokes方程真空状态的动力学行为

本文主要考虑了一维可压Navier-Stokes方程真空状态的动力学行为.对于任意的熵弱解,如果初始状态不存在真空,我们证明了密度函数关于时间和空间变量是连续的且对于任意时间它是处处为正的.同时,我们还得到了含有间断连接的真空状态的整体熵弱解的存在性,结果显示其真空区域以代数速率被压缩,并在有限时间内消失.     

A characterization of parallelepipeds related to weak derivatives

Summary We characterize parallelepipeds in R^m within the family of all convex bodies by a property of special measures on its boundary. We show that these measures are related to weak derivatives (in the sense of [5] and [8]) of convex-valued functions. The results can be applied (see [9]) to derive a generalization of a theorem of Lehmann (see [4]) on the comparison of uniform location experiments.     

多元凸函数的一个定理

本文在［１］的基础上．给出了与多元函数中值定理结论类似的一个定理．     

关于集值优化问题的若干结果

引入了集值集值C-τ-半预不变凸概念，证明了集值集值C-τ-半预不变凸优化问题的局部弱有效元是弱有效元，给出了集值预不变凸变分不等式作为集值C-τ-半预不变凸优化问题的充分条件和必要条件，这些结果推广了文[1-4]的相应结果。     

Invariant Measures for Set-Valued Dynamical Systems

A continuous map on a compact metric space, regarded as a dynamical system by iteration, admits invariant measures. For a closed relation on such a space, or, equivalently, an upper semicontinuous set-valued map, there are several concepts which extend this idea of invariance for a measure. We prove that four such are equivalent. In particular, such relation invariant measures arise as projections from shift invariant measures on the space of sample paths. There is a similarly close relationship between the ideas of chain recurrence for the set-valued system and for the shift on the sample path space.

     

推导刚体动力学方程的张量方法

本文提出了基于连续介质力学概念推导刚体动力学方程的张量方法,运用具有零共旋率的惯性张量的时间导数公式,证明了Lagrange方程、Nielsen方程、Gibbs-Appell方程、Kane方程和广义动量式Kane方程等五种方法的等价性,给出了角速度、角加速度之间的一些微分关系式.     

Besov estimates for weak solutions of a class of quasilinear parabolic equations with general growth

In this paper, we obtain the local higher fractional differentiability regularity estimates in Besov spaces of weak solutions for the quasilinear parabolic equations with general growth under some proper conditions on $a,A$, and $\mathbf{F}$. Moreover, we would like to point out that the results in the present work improve the known results for such parabolic equations.     

Mathematical Analysis of a Chemotaxis-Type Model of Soil Carbon Dynamic

The goal of this paper is to study the mathematical properties of a new model of soil carbon dynamics which is a reaction-diffusion system with a chemotactic term, with the aim to account for the formation of soil aggregations in the bacterial and microorganism spatial organization (hot spot in soil). This is a spatial and chemotactic version of MOMOS (Modelling Organic changes by Micro-Organisms of Soil), a model recently introduced by M. Pansu and his group. The authors present here two forms of chemotactic terms, first a “classical” one and second a function which prevents the overcrowding of microorganisms. They prove in each case the existence of a nonnegative global solution, and investigate its uniqueness and the existence of a global attractor for all the solutions.     

Behavior of Solutions to the Dirichlet Problem for Elliptic Systems in Convex Domains

We consider the Dirichlet problem for strongly elliptic systems of order 2m in convex domains. Under a positivity assumption on the Poisson kernel it is proved that the weak solution has bounded derivatives up to order m provided the outward unit normal has no big jumps on the boundary. In the case of second order symmetric systems in plane convex domains the boundedness of the first derivatives is proved without the assumption on the normal.     

A Set-Valued Analysis Approach to Second Order Differentiation of Nonsmooth Functions

A theory of second order differentiability for real functions with calm Clarke gradient at points of strict differentiability is presented from a set-valued analysis perspective by using the theory of graphical derivatives of set-valued maps. Connections with other second order theories and applications in optimization are given. This work is partially supported by Ministerio de Educación y Ciencia (Spain), project MTM2006-02629 and IngenioMathematica (i-MATH) CSD2006-00032 (Consolider-Ingenio 2010).     

Stability for Equilibrium Problems: From Variational Inequalities to Dynamical Systems

We study the connections between solutions of variational inequalities and equilibrium points of a generalized dynamical system. Furthermore, we analyze some stability questions arising in this field.     

A Categorical Approach to Threshold Agent Networks

Threshold agent networks (TANs) constitute a discretized modification of threshold (also known as neural) networks that are appropriate for modeling computer simulations. In this paper morphisms are introduced between TANs, thus forming a category TAN. Several of the properties of this category are explored. A generalization of TANs to allow more flexible interactions between the agents is then introduced. The new objects are also endowed with appropriate morphisms and the properties of the newly obtained category are compared with the properties of its older subcategory.     

Relation between Different Types of Global Attractors of Set-Valued Nonautonomous Dynamical Systems

The article is devoted to the study of the relation between forward and pullback attractors of set-valued nonautonomous dynamical systems (cocycles). Here it is proved that every compact global forward attractor is also a pullback attractor of the set-valued nonautonomous dynamical system. The inverse statement, generally speaking, is not true, but we prove that every global pullback attractor of an α-condensing set-valued cocycle is always a local forward attractor. The obtained general results are applied while studying periodic and homogeneous systems. We give also a new criterion of the absolute asymptotic stability of nonstationary discrete linear inclusions. Dedicated to our friend Professor Enrico Primo Tomasini on the occasion of his 55th birthdayMathematics Subject Classifications (2000) Primary: 34C35, 34D20, 34D40, 34D45, 58F10,58F12, 58F39; secondary: 35B35, 35B40.     

Rotating periodic solutions for second‐order dynamical systems with singularities of repulsive type

In this paper, we study the following second‐order dynamical system: where c ?0 is a constant, and . When g admits a singularity at zero of repulsive type without the restriction of strong force condition, we apply the coincidence degree theory to prove that the system admits nonplanar collisionless rotating periodic solutions taking the form u (t  + T ) = Q u (t ), with T  > 0 and Q an orthogonal matrix under the assumption of Landesman–Lazer type. Copyright © 2016 John Wiley & Sons, Ltd.     

Kovalevskaya Top: An Elementary Approach

We give an elementary, very short solution to the equations of motion for the Kovalevskaya top, using some results from the original papers by Kovalevskaya, Kötter, and Weber and also the Lax representation obtained by the author.     

Infinitely Many Noncollision Periodic Solutions for Planar Keplerian-Like 2-Body Problems

Using the implicit rotational invariant property for the potential and the variational functional, we prove the existence of infinitely many noncollision periodic solutions for planar Keplerian-like 2-body problems.AMS Mathematics Subject Classification: 34C15, 34C25, 58F, 70F10