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Klappentext 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 Savar 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 surveyof 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 Zusammenfassung Talks about partial differential equations! ranging from the abstract theory of gradient flows to stochastic representations of non-linear parabolic PDE's. This work considers more concrete probelms relating to specific applications. Inhaltsverzeichnis Preface Contributors 1.L. Ambriosio, G. Savaré: Gradient flows of probability measures 2.M.A. Herrero: The mathematics of chemotaxis 3.N. Masmoudi: Examples of singular limits in hydrodynamics 4. C. DeLellis: Notes on hyperbolic systems of conservation laws and transport equations 5. G. Rein: Collisionless kinetic equations from astrophysics - the Vlasov-Poisson system 6. H.M. Stochastic representations for non-linear parabolic PDE's 7. E. Zuazua Controllability and observability of partial differential equations: Some results and open problems Index ...
