The Rein Equation

The material collected in this volume reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear parabolic PDE's. Articles will highlight the present as well as expected future directions of development of the field with particular emphasis on applications. The article by Ambrosio and Savaré discusses the most recent development in the theory of gradient flow of probability measures. After an introduction reviewing the properties of the Wasserstein space and corresponding subdifferential calculus, applications are given to evolutionary partial differential equations. The contribution of Herrero provides a description of some mathematical approaches developed to account for quantitative as well as qualitative aspects of chemotaxis. Particular attention is paid to the limits of cell's capability to measure external cues on the one hand, and to provide an overall description of aggregation models for the slim mold Dictyostelium discoideum on the other. The chapter written by Masmoudi deals with a rather different topic - examples of singular limits in hydrodynamics. This is nowadays a well-studied issue given the amount of new results based on the development of the existence theory for rather general systems of equations in hydrodynamics. The paper by DeLellis addreses the most recent results for the transport equations with regard to possible applications in the theory of hyperbolic systems of conservation laws. Emphasis is put on the development of the theory in the case when the governing field is only a BV function. The chapter by Rein represents a comprehensive survey of results on the Poisson-Vlasov system in astrophysics. The question of global stability of steady states is addressed in detail. The contribution of Soner is devoted to different representations of non-linear parabolic equations in terms of Markov processes. After a brief introduction on the linear theory, a class of non-linear equations is investigated, with applications to stochastic control and differential games. The chapter written by Zuazua presents some of the recent progresses done on the problem of controllabilty of partial differential equations. The applications include the linear wave and heat equations,parabolic equations with coefficients of low regularity, and some fluid-structure interaction models. - Volume 1 focuses on the abstract theory of evolution - Volume 2 considers more concrete probelms relating to specific applications - Volume 3 reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear PDEs

Accompanying DVD-ROM contains the electronic proceedings of the summer school on mathematical general relativity and global properties of solutions of Einstein's equations held at Cargèse, Corsica, France, July 20-Aug. 10, 2002.

Earth reinforcing techniques are increasingly becoming a useful, powerful and economical solution to various problems encountered in geotechnical engineering practice. Expansion of the experiences and knowledge in this area has succeeded in developing new techniques and their applications to geotechnical engineering problems. In order to discuss the latest experiences and knowledge, and with the purpose of spreading them all over the world for further development, the IS Kyushi conference series on the subject of earth reinforcement have been held in Fukuoka, Japan, every four years since 1988. This fourth symposium, entitled Landmarks in Earth Reinforcement, is a continuation of the series IS Kyushu conferences, and also aims at being one of the landmarks in the progress of modern earth reinforcement practice. The first volume contains 137 papers selected for the symposium covering almost every aspect of earth reinforcement. The second volume contains texts of the special and keynote lectures.

Earth reinforcement techniques are used worldwide, providing dependable solutions to a wide range of geotechnical engineering problems.Well-established earth reinforcement technologies are regularly augmented by new materials, innovative construction techniques and advances in design and analysis. Furthermore, reinforced earth structures are increasingly seen as expedient and economical techniques in disaster situations, such as earthquakes, flooding or tsunamis. NEW HORIZONS in EARTH REINFORCEMENT contains contributions from the 5th International Symposium on Earth Reinforcement, Kyushu, Japan, 14-16 November 2007, and presents the very latest earth reinforcement techniques and design procedures. The volume showcases advances in materials and emerging applications, with special emphasis on disaster mitigation and geoenvironmental issues. The book will be invaluable to academics and professionals in geotechnical engineering.