Vladimir Gol'dshtein

Axiomatic Theory of Sobolev Spaces

We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples (Hajtasz Sobolev spaces, weighted Sobolev spaces, Upper-gradients, etc). We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic and p-hyperbolic spaces are then discussed.

Simple global reduction technique based on decomposition approach

Large and complex (nonlinear) models of chemical kinetics are one of the major obstacles in simulations of reacting flows. In the present work a new approach for an automatic reduction of chemical kinetics models, the so-called Global Quasi-Linearization (GQL) method is presented. The method is similar to the ILDM and CSP approaches in the sense that it is based on a decomposition into fast/slow motions and on slow invariant manifolds, but has a global character which allows us to overcome difficulties with the application of slow invariant manifolds and significantly simplifies the construction procedure for approximation of the slow invariant system manifold. The method is implemented within the standard ILDM method and applied to a number of model examples and to a meaningful combustion chemistry model.

Criterion for Thermal Explosion with Reactant Consumption in a Dusty Gas

The thermal explosion problem in a dusty gas is investigated. Dynamical regimes of the system are classified: slow regimes, thermal explosion with delay, thermal explosion (without delay). The critical transition conditions for the different dynamical regimes are analysed. We emphasize that the critical condition for transition between slow regimes and explosion with delay is a thermal explosion limit. The thermal explosion limit is described in the phase space by a so-called duck-trajectory. (The notion of duck-trajectory was introduced by Cartier for detailed investigation of relaxation oscillations.)

Structure of the thermal explosion limit

Abstract In a closed system an explosion limit is the transition region which separates the regions of slow and explosive reactions with respect to initial conditions. In the present paper the characteristic features of chemical reaction in the transition region are investigated for thermal explosion. The critical phase trajectory of the system contains an unstable integral manifold. Initial conditions corresponding to this trajectory can be understood as the explosion limit. An equation determining maximum temperature rise on the critical trajectory is obtained. The transition region can be divided into regions of slow and explosive transient regimes. The transition region is characterized by the extrema of induction period and degree of reactant consumption which correspond to a maximum reaction rate.

Thermal radiation effect on thermal explosion in gas containing fuel droplets

The effect of thermal radiation on the dynamics of a thermal explosion of a flammable gas mixture with the addition of volatile fuel droplets is studied. This is based on an original physical model of self-ignition. The thermal radiation energy exchange between the evaporating surface of the fuel droplets and burning gas is described using the P-1 model with Marshak boundary conditions. The original system of equations describing the effects of heating, evaporation and the combustion of fuel droplets is simplified to enable their analysis using asymptotic methods. The mathematical formulation is eventually reduced to a singularly perturbed system of ordinary differential equations. This allows us to apply the advanced geometric asymptotic technique (integral manifold method) for the qualitative analysis of the behaviour of the solution. Possible types of dynamic behaviour of the system are classified and parametric regions of their existence are determined analytically. The main attention is concentrated ...

On combustion waves driven by diffusion of pressure

In gas filled porous media the local elevation of pressure slowly diffuses to the adjacent layers of gas inducing the rise in temperature there. In the case of explosive gases this mechanism may lead to the formation of a self-sustaining combustion wave propagating at a constant speed. It is argued that the barodiffusion may be responsible for the occurrence of the so-called high velocity regime often observed in filtration combustion It is shown that the high velocity regime may emanate from the low velocity regime controlled by the sysiems thermal diffusivity. It is suggested that the effect may be related to the classical problem of deflagration-to-detonation transition in narrow pipes.

Applications of change of variables operators for exact embedding theorems

We propose here a new method for the investigation of embedding operators. It is based on an exact description of classes of homeomorphisms that induce change of variables operators on the Sobolev spaces. This method permits to obtain new exact results about embedding operatorsWp1(G)↪Lq(G) (orC(G)) in domainsG ⊂ ℝn with irregular boundaries. For these types of domains the behavior of the limiting embedding exponentq1,G*(p) is very complicated and surprising. We construct classes of simply connected domains with nonsmooth functionq1,G*(p) and a class of multiply connected domains with discontinuous functionq1,G*(P).

Critical Conditions for Thermal Explosion with Reactant Consumption

Abstract The critical value of the dimensionless heat loss parameter is determined for a thermal explosion with reactant consumption. The critical phase trajectory of the system contains an unstable integral manifold as its own portion. Initial conditions corresponding to this trajectory conform to the explosion limit. The technique of asymptotic expansions is used to derive a critical explosion parameter

On thermal explosion of a cool spray in a hot gas

The effect of a flammable spray on thermal explosion in a preheated combustible gas mixture is investigated using a simplified model that contains the essentials of the basic physical processes at work. The study represents a re-examination of the question of the ignition of a spray of droplets from the viewpoint of an explosion problem, in which the droplets are taken to be a source of endothermicity. Use is made of various methods for the qualitative analysis of systems of differential equations in order to examine the dynamics of the system. Possible types of dynamical behavior of the system are looked into and parametric regions of their existence are determined analytically. Peculiarities, of these dynamical regimes are investigated, and their dependence on the physical system parameters are analyzed. In particular, analytical formulas are developed for ignition delay times by exploiting the sensitivity of the process to the chemical activation energy. A qualitative comparison of predicted ignition times with independent experimental measurements from the literature yields good order of magnitude agreement.

The Kelvin-Nevanlinna-Royden criterion for

Abstract. We generalize the so called Kelvin–Nevanlinna–Royden criterion for the parabolicity of manifolds to the case of p-parabolicity for all