HANDBOOK OF DIFFERENTIAL EQUATIONS: EVOLUTIONARY EQUATIONS, 3
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Edited By C.M. Dafermos, Brown University, Providence, RI, USA Eduard Feireisl, Mathematical Institute AS CR, Prague, Czech Republic.
Description 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 Savare 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.
Audience
University libraries and Research mathematicians
Contents Preface
Contributors
1.L. Ambriosio, G. Savare: 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
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