Well-posed model system

If the characteristics of the system of equations are found to be complex, the initial-value problem is said to be ill-posed [178]. A physical interpretation of this mathematical statement can be found by analyzing the flow instabilities predicted by this set of model equations. The instabilities predicted by a well-posed model system has some realistic physical meaning, while the instability always present in an ill-posed system is a mathematical mode having no physical origin indicating that the model is not treating small-scale phenomena correctly. 

It is not clear how improvements can be made without real progress on the difficult fundamental problems of diffusion in media with obstacles and cooperation of large-scale motions between interpenetrating chains which do not violate chain connectivity. The DeGennes reptation model (225) makes a significant contribution to the first problem, although in an admittedly simplified system. Rigorous calculations or computer simulations on well-defined models which relate to the second problem would be extremely valuable, even if the models themselves were not completely faithful representations of the assumed physical situation. It is not obvious how even to pose solvable problems, simplified or not, which relate to interchain cooperation. 

Like all other models, LES requires the specification of proper boundary and initial conditions in order to fully determine the system and obtain a mathematically well-posed problem. However, this concept deviates from the more familiar average models in that the boundary conditions apparently rep>-resent the whole fluid domain beyond the computational domain. Therefore, to specify the solution completely, these conditions must apply to all of the space-time modes it comprises. 

The well-posedness of the two-fluid model has been a source of controversy reflected by the large number of papers on this issue that can be found in the literature. This issue is linked with analysis of the characteristics, stability and wavelength phenomena in multi -phase flow equation systems. The controversy originates primarily from the fact that with the present level of knowledge, there is no general way to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem. The mathematical theory of well posedness for systems of partial differential equations describing dispersed chemical reacting flows needs to be examined. 

The above description is formal and although the rate calculation is well-posed it is difficult to carry out for problems of chemical interest. Hence, much of the focus of recent theoretical studies of reaction rates in the condensed phases has centered on evaluation of the reaction rate for model systems [8]. Perhaps the best way to obtain information on the structure of the rate coefficient expressions is through direct molecular dynamics simulations on model systems. Since both of these approaches have been reviewed by others in this volume, we now turn our attention to far-from-equilibrium systems where some of the ideas summarized above can be used, but new phenomena appear which require special techniques. 

Geochemists, however, seem to have reached a consensus (e.g., Karpov and Kaz min, 1972 Morel and Morgan, 1972 Crerar, 1975 Reed, 1982 Wolery, 1983) that Newton-Raphson iteration is the most powerful and reliable approach, especially in systems where mass is distributed over minerals as well as dissolved species. In this chapter, we consider the special difficulties posed by the nonlinear forms of the governing equations and discuss how the Newton-Raphson method can be used in geochemical modeling to solve the equations rapidly and reliably. 

The experienced catalytic chemist or chemical reaction engineer will immediately recognize that the study of a new catalytic reaction system using an in situ spectroscopy, has a great deal in common with the concepts of inverse problems and system identification. First, there is a physical system which cannot be physically disassembled, and the researcher seeks to identify a model for the chemistry involved. The inverse in situ spectroscopic problem can be denoted by Eq. (2). Secondly, the physical system evolves in time and spectroscopic measurements as a function of time are a must. There are realistic limitations to the spectroscopic measurements performed. For this reason as well as for various other reasons, the inverse problem is ill-posed (see Section 4.3.6). Third, signal processing will be needed to filter and correct the raw data, and to obtain a model of the system. The ability to have the individual pure component spectra of the species present in... 

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