Dynamics of internal layers and diffusive interfaces.

*(English)*Zbl 0684.35001
CBMS-NSF Regional Conference Series in Applied Mathematics 53. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 0-89871-225-4). vi, 93 p. (1988).

Interfacial phenomena are met when a continuum is present which can exist in two different chemical or physical “states” and a mechanism provides a spatial separation between these states. For example the separation boundary, i.e., the interface, may result from the interplay of a diffusive effect that attempts to mix the material in its different states and hence to “smooth” their properties and another “unmixing” tendency that works to drive it to the one or the other pure state. Five cases of such physical or chemical mechanisms producing an unmixing effect are studied in this work. Although the differences between the respective phenomena are significant and important there exists a body of mathematical concepts and techniques which serves as a ground for the mathematical analysis of a variety of types and properties of interfaces. The presentation starts with a survey of this common mathematical basis and has the aim to develop and study the particular properties, phenomena and problems related to the different mechanisms.

Assume there is a dynamical process from which evolves a smooth state variable, i.e., a function of time, two dimensional space and a small positive parameter. Those situations are of interest in which the process generates and preserves a moving internal layer, the study focusses on the dynamics of a fully developed layer. The first paragraph consists of an asymptotic investigation of u with respect to the layer and a discussion of the respective approximations. Three models serve as examples for the application of these basic mathematical tools: An equation of bistable type that has been used to describe waves in population genetics, physiology and nonlinear transmission lines, a system of two equations for the temperature and a “phase function” that represents a phase field model, and a system of conservation laws with small dissipation where the separation is due to a diffusive shock layer.

Flame theory is studied in the second chapter. A certain emphasis lies on the detailed presentation of the stability analysis of the planar interface. The study is performed for a one step reaction in the framework of the thermodiffusive model which is constructed in the context of the basic body of asymptotics. Propagation of flames is due to production and diffusion of heat. Here, another contributing source is taken in consideration, the production of radicals in the combustion zone. Finally, the complexity of chemical processes involved in flame phenomena is studied in view of the goal to systematically obtain simpler models that provide the classical features of the flame in an easier understandable manner.

Models for the separation of ions due to an electric field and the related problems are discussed in a short chapter on electrophoresis. The mechanisms of isotachophoresis and isoelectric focussing are addressed.

Waves in excitable and self-oscillatory media are studied in the final chapter. This includes models for the propagation of signals along a nerve axon or cardiac tissue as well as such ones for the Belousov- Zhabotinskii or reactions of other excitable chemical reagents. The discussion includes topics as the correction to the velocity due to curvature, corner layers and the twisting action. The last section reports on structures as rotating spirals and expanding rings.

The work is completed by a comprehensive up-to-date bibliography.

Nonlinear waves is a highly interesting rapidly developing field in mathematics and its applications. This is the fascinating state-of-art survey of one of the main contributors to the far from complete theory.

Assume there is a dynamical process from which evolves a smooth state variable, i.e., a function of time, two dimensional space and a small positive parameter. Those situations are of interest in which the process generates and preserves a moving internal layer, the study focusses on the dynamics of a fully developed layer. The first paragraph consists of an asymptotic investigation of u with respect to the layer and a discussion of the respective approximations. Three models serve as examples for the application of these basic mathematical tools: An equation of bistable type that has been used to describe waves in population genetics, physiology and nonlinear transmission lines, a system of two equations for the temperature and a “phase function” that represents a phase field model, and a system of conservation laws with small dissipation where the separation is due to a diffusive shock layer.

Flame theory is studied in the second chapter. A certain emphasis lies on the detailed presentation of the stability analysis of the planar interface. The study is performed for a one step reaction in the framework of the thermodiffusive model which is constructed in the context of the basic body of asymptotics. Propagation of flames is due to production and diffusion of heat. Here, another contributing source is taken in consideration, the production of radicals in the combustion zone. Finally, the complexity of chemical processes involved in flame phenomena is studied in view of the goal to systematically obtain simpler models that provide the classical features of the flame in an easier understandable manner.

Models for the separation of ions due to an electric field and the related problems are discussed in a short chapter on electrophoresis. The mechanisms of isotachophoresis and isoelectric focussing are addressed.

Waves in excitable and self-oscillatory media are studied in the final chapter. This includes models for the propagation of signals along a nerve axon or cardiac tissue as well as such ones for the Belousov- Zhabotinskii or reactions of other excitable chemical reagents. The discussion includes topics as the correction to the velocity due to curvature, corner layers and the twisting action. The last section reports on structures as rotating spirals and expanding rings.

The work is completed by a comprehensive up-to-date bibliography.

Nonlinear waves is a highly interesting rapidly developing field in mathematics and its applications. This is the fascinating state-of-art survey of one of the main contributors to the far from complete theory.

Reviewer: H.Jeggle

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35K55 | Nonlinear parabolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

92D25 | Population dynamics (general) |

80A30 | Chemical kinetics in thermodynamics and heat transfer |

92Cxx | Physiological, cellular and medical topics |

92Exx | Chemistry |

80A32 | Chemically reacting flows |

80A17 | Thermodynamics of continua |

76T99 | Multiphase and multicomponent flows |

76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |

35B20 | Perturbations in context of PDEs |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35B35 | Stability in context of PDEs |

35C20 | Asymptotic expansions of solutions to PDEs |

35C10 | Series solutions to PDEs |

35K40 | Second-order parabolic systems |

35K57 | Reaction-diffusion equations |

78A25 | Electromagnetic theory (general) |