Handbook of Differential Equations: Evolutionary Equations

The stuff is in stochastic representations of PDE's non-linear parabolic.
Articles will highlight the future with an emphasis on applications.
The article by Ambrosio and Savaré discusses
the most recent development in the theory of the 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 quantitative description of some chemotaxis. Particular attention is paid to the limits of cell's
capability to measure external cues on the one hand. Dictyostelium discoideum on the other.
The chapter written by Masmoudi deals with hydrodynamics. This is nowadays a well-explaining theory of hydrodynamics. The paper by DeLellis addreses on the theory of hyperbolic systems. Emphasis is put on the development of the function of the BV.
The chapter by Rein represents a comprehensive survey of 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 to 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 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 medicines more concrete probelms relating to specific applications
- Volume 3 reflects the abstract presentations of stochastic representations of non-linear PDEs

Preface

Contributors

1.L. Ambriosio, G. Savaré: Gradient flows of probability measures

2.MA Herrero: The mathematics of chemotaxis

3.N. Masmoudi: Examples of singular limits in hydrodynamics

4. C. DeLellis: Notes on hyperbolic systems of conservation and transport equations

5. G. Rein: Collisionless kinetic equations from astrophysics - the Vlasov-Poisson system

6. HM Stochastic representations for non-linear parabolic PDE's

7. E. Zuazua. Controllability and observation of partial differential equations

index