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ISBN: 9783030321611 Year: 2020 Publisher: Cham Springer International Publishing :Imprint: Springer
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ISBN: 9783540737056 Year: 2008 Publisher: Berlin Heidelberg Springer Berlin Heidelberg
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Entropy and entropy production have recently become mathematical tools for kinetic and hydrodynamic limits, when deriving the macroscopic behaviour of systems from the interaction dynamics of their many microscopic elementary constituents at the atomic or molecular level. During a special semester on Hydrodynamic Limits at the Centre Émile Borel in Paris, 2001 two of the research courses were held by C. Villani and F. Rezakhanlou. Both illustrate the major role of entropy and entropy production in a mutual and complementary manner and have been written up and updated for joint publication. Villani describes the mathematical theory of convergence to equilibrium for the Boltzmann equation and its relation to various problems and fields, including information theory, logarithmic Sobolev inequalities and fluid mechanics. Rezakhanlou discusses four conjectures for the kinetic behaviour of the hard sphere models and formulates four stochastic variations of this model, also reviewing known results for these.
Partial differential equations --- Operational research. Game theory --- Mathematical physics --- differentiaalvergelijkingen --- stochastische analyse --- wiskunde --- fysica --- kansrekening
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ISBN: 9783642218613 Year: 2011 Publisher: Berlin Heidelberg Springer Berlin Heidelberg Imprint Springer
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This volume collects the notes of the CIME course "Nonlinear PDE's and applications" held in Cetraro (Italy) on June 23-28, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They presented a broad overview of far-reaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, Hamilton-Jacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rate-independent evolution problems. The book explores the fundamental connections between all of these topics and points to new research directions in contributions by leading experts in these fields.
Differential geometry. Global analysis --- Functional analysis --- Partial differential equations --- Mathematical analysis --- Numerical methods of optimisation --- differentiaalvergelijkingen --- analyse (wiskunde) --- differentiaal geometrie --- functies (wiskunde) --- kansrekening --- optimalisatie
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