Please fill in your query. A complete syntax description you will find on the General Help page.
Mathematical methods for hydrodynamic limits. (English)
Lecture Notes in Mathematics. 1501. Berlin: Springer-Verlag. (1991).
Considerable progress has been made in the last years concerning the rigorous derivation of hydrodynamical behaviour of microscopic models. Though these models are unrealistic from a physical point of view, they contribute to the general understanding and may lead to more realistic ones. This explains, why soon after the appearance of {\it H. Spohn’s} book [Large scale dynamics of interacting particles (1991; Zbl 0742.76002)] here are lecture notes treating the same subject. Compared to Spohn’s comprehensive survey the authors aim “to present some of the mathematical methods which are more often used by examining in a self- contained way particular models”. After an introduction, in Chapters II to VI macroscopic dynamics are derived for several microscopic models. Independent particles serve to present the basic concepts. The method of entropy estimates introduced by Guo, Papanicolaou and Varadhan are demonstrated by the zero range process. Most models (microscopic models for reaction diffusion equations and for the Carleman equation and the Glauber and Kawasaki process) are treated with correlation function methods. Chapters VII to X deal with long time behaviour and critical phenomena. The first two of them discuss in an informal way without proofs the transition from behaviour described by kinetic equations to Euler and Navier-Stokes type equations for long times resp. phase separation and interface dynamics. The last two chapters study in detail the transition from an unstable equilibrium to one of the two pure phases for a Glauber and Kawasaki process. The proof is based on the use of correlation functions, too. In the sense of their aim mentioned above, the authors present interesting mathematical models and methods for important physical phenomena. Their lecture notes can be recommended to anybody interested in such problems. One incidental remark: the title may be misleading, since except for independent particles and the zero range process the treated limits are not hydrodynamical in the strict sense with exclusive rescaling of time and space (see also Def. 2.5.4), but in addition include a rescaling of the dynamics.
M.Mürmann (Heidelberg)
Valid XHTML 1.0 Transitional Valid CSS!