Handbook of Differential Equations Evolutionary Equations Volume 4 1st Edition by Constantin M Dafermos, Milan Pokorny ISBN 0080931979 9780444530349
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ISBN 10: 0080931979
ISBN 13: 9780444530349
Author: Constantin M Dafermos, Milan Pokorny
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 Savaré 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.
– Volume 1 focuses on the abstract theory of evolution
– Volume 2 considers more concrete probelms relating to specific applications
– Volume 3 reflects the active present of this area of mathematics, ranging from the abstract theory of gradient flows to stochastic representations of non-linear PDEs
Handbook of Differential Equations Evolutionary Equations Volume 4 1st Table of contents:
Chapter 1: Gradient Flows of Probability Measures
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Introduction
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1 Notation and measure-theoretic results
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2 Metric and differentiable structure of the Wasserstein space
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3 Convex functionals in P2(Rd)mathcal{P}_2(mathbb{R}^d)P2(Rd)
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4 Subdifferential calculus in P2(Rd)mathcal{P}_2(mathbb{R}^d)P2(Rd)
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5 Gradient flows of λlambdaλ-geodesically convex functionals in P2(Rd)mathcal{P}_2(mathbb{R}^d)P2(Rd)
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6 Applications to evolution PDEs
Chapter 2: The Mathematics of Chemotaxis
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Abstract
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1 Introduction: What is chemotaxis?
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2 How do chemotactic units work?
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3 Some mathematical problems arising from the study of Dictyostelium discoideum
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Acknowledgements
Chapter 3: Examples of Singular Limits in Hydrodynamics
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Abstract
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1 Introduction
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2 The inviscid limit
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3 Compressible–incompressible limit
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4 Study of rotating fluids at high frequency
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5 Hydrodynamic limit of the Boltzmann equation
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6 Some homogenization problems
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7 Conclusion
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Acknowledgment
Chapter 4: Notes on Hyperbolic Systems of Conservation Laws and Transport Equations
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1 Introduction
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2 Preliminaries
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3 DiPerna–Lions theory for nearly incompressible flows
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4 Commutator estimates and Ambrosio’s renormalization theorem
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5 Existence, uniqueness and stability for the Keyfitz and Kranzer system
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6 Blow-up of the BV norm for the Keyfitz and Kranzer system
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7 Partial regularity and trace properties of solutions to transport equations
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8 Bressan’s compactness conjecture and the renormalization conjecture for nearly incompressible BV vector fields
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9 Tangential sets of BV vector fields
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Acknowledgements
Chapter 5: Collisionless Kinetic Equations from Astrophysics – The Vlasov–Poisson System
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Introduction
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Notation and preliminaries
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1 Classical solutions to the initial value problem
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2 Stability
Chapter 6: Stochastic Representations for Nonlinear Parabolic PDEs
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Abstract
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1 Introduction
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2 Linear case: Feynman–Kac representation
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3 Representation via controlled processes
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4 Backward representations
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5 Monte Carlo methods
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Acknowledgement
Chapter 7: Controllability and Observability of Partial Differential Equations: Some Results and Open Problems
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Abstract
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1 Introduction
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2 Preliminaries on finite-dimensional systems
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3 Controllability of the linear wave equation
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4 The heat equation
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5 Sharp observability estimates for the linear heat equation
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6 Parabolic equations with low regularity coefficient
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7 Fluid-structure interaction models
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8 Some open problems
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Tags: Constantin M Dafermos, Milan Pokorny, Differential Equations, Evolutionary Equations