ILL-POSEDNESS FOR THE NONLINEAR DAVEY-STEWARTSON EQUATION

The nonlinear D-S equations on R^d, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space H^s whenever the exponent s is lower than that predicted by scaling or Galilean invariance, or when the regularity is too low to support distributional solutions. Authors analyze a class of solutions for which the zero-dispersion limit provides good approximations.     

Distance to Ill-Posedness in Linear Optimization via the Fenchel-Legendre Conjugate

We consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space and with a fixed index set, endowed with the topology of the uniform convergence of the coefficient vectors. A system is ill-posed with respect to the consistency when arbitrarily small perturbations yield both consistent and inconsistent systems. In this paper, we establish a formula for measuring the distance from the nominal system to the set of ill-posed systems. To this aim, we use the Fenchel-Legendre conjugation theory and prove a refinement of the formula in Ref. 1 for the distance from any point to the boundary of a convex set.This research has been partially supported by grants BFM2002–04114-C02 (01–02) from MEC (Spain) and FEDER (EU) and by grants GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain).     

Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model

In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.

     

On well-posedness of the dynamic problem for an anisotropic fluid-saturated solid

It is known that a high degree of anisotropy in the constitutive behaviour of a solid may result in the loss of hyperbolicity of the dynamic equations in the form of either complex-conjugate or purely imaginary characteristic wave speeds (flutter ill-posedness and shear band formation, respectively). In the present paper we investigate the characteristic wave speeds in the dynamic problem for a transversely isotropic fluid-saturated porous solid. Three cases are considered: a dry solid and a saturated solid under locally undrained and drained conditions. It is shown that, for given constitutive parameters of the solid skeleton, the dynamic problem for a drained solid may become ill-posed due to the flutter-type loss of hyperbolicity, while the dynamic equations for a dry and an undrained solids remain hyperbolic. For a given solid skeleton, the characteristic wave speeds are strongly influenced by the pore fluid compressibility which, in turn, is extremely sensitive to the presence of a small amount of free gas.     

Well- and Ill-Posedness Issues for a Class of 2D Wave Equation with Exponential Growth

Extending the previous work [1], we establish well-posedness results for a more general class of semilinear wave equations with exponential growth. First, we investigate the well-posedness in the energy space. Then, we prove the propagation of the regularity in the Sobolev spaces HS（IR^2） with s 〉 1. Finally, an ill-posedness result is obtained in HS（IR^2） for s 〈 1.     

A Sharp Bilinear Estimate for the Bourgain-Type Space with Application to the Benjamin Equation

This note shows the existence of a sharp bilinear estimate for the Bourgain-type space and gives its application to the optimal local well/ill-posedness of the Cauchy problem for the Benjamin equation.     

弹性力学非线性反演方法概述

考虑正演模型和观测数据确定性性质的不同, 对弹性力学反问题解的定义进行了归纳总结. 非线性和不适定性是反问题理论研究和工程应用的瓶颈问题, 对反问题非线性和不适定性的有效处理方法进行了总结.并在此基础上介绍了非线性反演问题求解方法的一些最新进展, 对弹性力学非线性反演方法进行了归纳分类.     

Acceleration wave speeds in a hypoplastic constitutive model

The paper presents a theoretical investigation of acceleration waves in a plastic material described by an incrementally non-linear hypoplastic constitutive equation. Speeds of plane acceleration waves and their dependence on the stress state are calculated. The spectrum of possible wave speeds is found to be continuous, which is in contrast to discrete wave speed spectra in incrementally linear models. Two types of ill-posedness are revealed, known as flutter ill-posedness and stationary discontinuity. The wave speed analysis is also performed by the characteristic method, leading to different equations compared to the acceleration wave approach. It is proved that for the considered hypoplastic constitutive equation both approaches give identical wave speed spectra.