Solving Strategies and Well-Posedness in Linear Semi-Infinite Programming

In this paper we introduce the concept of solving strategy for a linear semi-infinite programming problem, whose index set is arbitrary and whose coefficient functions have no special property at all. In particular, we consider two strategies which either approximately solve or exactly solve the approximating problems, respectively. Our principal aim is to establish a global framework to cope with different concepts of well-posedness spread out in the literature. Any concept of well-posedness should entail different properties of these strategies, even in the case that we are not assuming the boundedness of the optimal set. In the paper we consider three desirable properties, leading to an exhaustive study of them in relation to both strategies. The more significant results are summarized in a table, which allows us to show the double goal of the paper. On the one hand, we characterize the main features of each strategy, in terms of certain stability properties (lower and upper semicontinuity) of the feasible set mapping, optimal value function and optimal set mapping. On the other hand, and associated with some cells of the table, we recognize different notions of Hadamard well-posedness. We also provide an application to the analysis of the Hadamard well-posedness for a linear semi-infinite formulation of the Lagrangian dual of a nonlinear programming problem.     

Cauchy-type problem for an abstract differential equation with fractional derivatives

The uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator A is proved. For an unbounded operator A we present a test for the uniform well-posedness of the problem under consideration consistent with the test for the uniform well-posedness of the Cauchy problem for an equation of second order.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 28–41.Original Russian Text Copyright © 2005 by A. V. Glushak.This revised version was published online in April 2005 with a corrected issue number.     

Remark on well-posedness for the fourth order nonlinear Schrödinger type equation

We consider the initial value problem for the fourth order nonlinear Schrödinger type equation (4NLS) related to the theory of vortex filament. In this paper we prove the time local well-posedness for (4NLS) in the Sobolev space, which is an improvement of our previous paper.

     

Staircase Parameterization in Dynamical Selection

This paper deals with a variant of a dynamical selection scheme introduced by Attouch and Cominetti for ill-posed convex minimization which combines approximation with the steepest descent method by mean of a suitable parameterization of the approximation parameter as a function of the time. This variant applies to a general inclusion with a maximal monotone operator by mean of a staircase parameterization. A discrete analogue is also considered. Applications to selecting a particular zero of a maximal monotone operator or a particular fixed point of a nonexpansive mapping via regularization techniques are presented. Finally, the alternative use of well-posedness by perturbations is discussed.     

一类广义层流方程组的初值问题的局部适定性

讨论层流理论中描述比重相近的层流间相互作用的方程组的广义方程组．通过引进一类函数空间和一系列先验估计,应用不动点原理,分别得到该类广义层流方程组的初值问题在H^s(R),s≥1的局部适定性．     

Well-posedness,Decay Estimates and Blow-up Theorem for the Forced NLS

In this article we prove that the following NLS iu_t = u_{zz}-g|u|^{P-1}u,g > O, x, t > 0 with either Dirichlet or Robin boundary condition at x = 0 is well-posed. L^{p + 1} decay estimates, blow-up theorem and numerical results are also given.     

一类大气演化方程组的Cauchy问题

在分层理论的框架下讨论一类大气演化方程组的Cauchy问题,证明了:1) 惯性力对一类大气演化方程组Cauchy问题的适定性判别标准没有影响;2) 可压缩性对粘性大气方程组Cauchy问题的适定性判别标准没有影响,但对无粘大气方程组,可压缩性改变Cauchy问题适定性判别标准;3) 所论方程组在t=0超平面上的Cauchy问题均是不适定的,并不受粘性和可压缩性的影响;4) 可压无粘大气方程与运动静止初始条件构成的Cauchy问题是不适定的．     

WELL-POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INVISCID COMPRESSIBLE ADIABATIC FLUID ＊

Introduction Itisreasonabletoconsiderarealfluidasanidealoneinfluidmechanicsundermany conditions.Forinstance,ingeneral,forthedistributionofthefluidfieldaroundtheaerocraft,mostpartofthefluidfieldmayberegardedasidealfluidexceptforasmallpartwhereeffects ofviscousandheatconductioninthethinlayernearthesurfacemustbeconsidered.Evenif thefluidiscompletelysupposedasidealonethroughoutthefluidfield,thequitereasonable resultsarealsogained,thereforestudyingidealfluidhasnotonlytheoreticalsignificancebut also…     

非静力旋转流体方程初值问题的适定性

应用分层理论所提供的方法,通过对非静力旋转流体方程这个数学模型本身的拓扑性质进行系统的理论分析和研究,来讨论对于这个方程的初值问题的几种不同提法以及相应的适定性问题．     

Integrability,existence of global solutions,and wave breaking criteria for a generalization of the Camassa–Holm equation

Recent generalizations of the Camassa–Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative of local solutions, for global well-posedness in Sobolev spaces for the family under consideration. Moreover, we prove that wave breaking phenomena occurs under certain mild hypothesis. Based on the machinery developed by Dubrovin [Commun. Math. Phys. 267, 117–139 (2006)] regarding bi-Hamiltonian deformations, we introduce the notion of quasi-integrability and prove that there exists a unique bi-Hamiltonian structure for the equation only when it is reduced to the Dullin–Gotwald–Holm equation. Our results suggest that a recent shallow water model incorporating Coriollis effects is integrable only in specific situations. Finally, to finish the scheme of geometric integrability of the family of equations initiated in a previous work, we prove that the Dullin–Gotwald–Holm equation describes pseudo-spherical surfaces.