Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations

Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations PDF

Author: Maria Colombo

Publisher: Springer

Published: 2017-06-07

Total Pages: 250

ISBN-13: 8876426078

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The first part of the book is devoted to the transport equation for a given vector field, exploiting the lagrangian structure of solutions. It also treats the regularity of solutions of some degenerate elliptic equations, which appear in the eulerian counterpart of some transport models with congestion. The second part of the book deals with the lagrangian structure of solutions of the Vlasov-Poisson system, which describes the evolution of a system of particles under the self-induced gravitational/electrostatic field, and the existence of solutions of the semigeostrophic system, used in meteorology to describe the motion of large-scale oceanic/atmospheric flows.​

Weighted Sobolev Spaces and Degenerate Elliptic Equations

Weighted Sobolev Spaces and Degenerate Elliptic Equations PDF

Author: Albo Carlos Cavalheiro

Publisher: Cambridge Scholars Publishing

Published: 2023-09-29

Total Pages: 333

ISBN-13: 1527551679

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In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is disturbed in the sense that some degeneration or singularity appears. This bad behavior can be caused by the coefficients of the corresponding differential operator as well as by the solution itself. There are several very concrete problems in various practices which lead to such differential equations, such as glaciology, non-Newtonian fluid mechanics, flows through porous media, differential geometry, celestial mechanics, climatology, and reaction-diffusion problems, among others. This book is based on research by the author on degenerate elliptic equations. This book will be a useful reference source for graduate students and researchers interested in differential equations.

Spaces of Measures and their Applications to Structured Population Models

Spaces of Measures and their Applications to Structured Population Models PDF

Author: Christian Düll

Publisher: Cambridge University Press

Published: 2021-10-07

Total Pages: 322

ISBN-13: 1009020471

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Structured population models are transport-type equations often applied to describe evolution of heterogeneous populations of biological cells, animals or humans, including phenomena such as crowd dynamics or pedestrian flows. This book introduces the mathematical underpinnings of these applications, providing a comprehensive analytical framework for structured population models in spaces of Radon measures. The unified approach allows for the study of transport processes on structures that are not vector spaces (such as traffic flow on graphs) and enables the analysis of the numerical algorithms used in applications. Presenting a coherent account of over a decade of research in the area, the text includes appendices outlining the necessary background material and discusses current trends in the theory, enabling graduate students to jump quickly into research.

An Introduction To The Geometrical Analysis Of Vector Fields: With Applications To Maximum Principles And Lie Groups

An Introduction To The Geometrical Analysis Of Vector Fields: With Applications To Maximum Principles And Lie Groups PDF

Author: Stefano Biagi

Publisher: World Scientific

Published: 2018-12-05

Total Pages: 450

ISBN-13: 9813276630

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This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings:

Time-Varying Vector Fields and Their Flows

Time-Varying Vector Fields and Their Flows PDF

Author: Saber Jafarpour

Publisher: Springer

Published: 2014-10-10

Total Pages: 119

ISBN-13: 3319101390

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This short book provides a comprehensive and unified treatment of time-varying vector fields under a variety of regularity hypotheses, namely finitely differentiable, Lipschitz, smooth, holomorphic, and real analytic. The presentation of this material in the real analytic setting is new, as is the manner in which the various hypotheses are unified using functional analysis. Indeed, a major contribution of the book is the coherent development of locally convex topologies for the space of real analytic sections of a vector bundle, and the development of this in a manner that relates easily to classically known topologies in, for example, the finitely differentiable and smooth cases. The tools used in this development will be of use to researchers in the area of geometric functional analysis.

Fokker–Planck–Kolmogorov Equations

Fokker–Planck–Kolmogorov Equations PDF

Author: Vladimir I. Bogachev

Publisher: American Mathematical Society

Published: 2022-02-10

Total Pages: 495

ISBN-13: 1470470098

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This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker–Planck–Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.

Smooth Ergodic Theory of Random Dynamical Systems

Smooth Ergodic Theory of Random Dynamical Systems PDF

Author: Pei-Dong Liu

Publisher: Springer

Published: 2006-11-14

Total Pages: 233

ISBN-13: 3540492917

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This book studies ergodic-theoretic aspects of random dynam- ical systems, i.e. of deterministic systems with noise. It aims to present a systematic treatment of a series of recent results concerning invariant measures, entropy and Lyapunov exponents of such systems, and can be viewed as an update of Kifer's book. An entropy formula of Pesin's type occupies the central part. The introduction of relation numbers (ch.2) is original and most methods involved in the book are canonical in dynamical systems or measure theory. The book is intended for people interested in noise-perturbed dynam- ical systems, and can pave the way to further study of the subject. Reasonable knowledge of differential geometry, measure theory, ergodic theory, dynamical systems and preferably random processes is assumed.

On First and Second Order Planar Elliptic Equations with Degeneracies

On First and Second Order Planar Elliptic Equations with Degeneracies PDF

Author: Abdelhamid Meziani

Publisher:

Published: 2011

Total Pages: 77

ISBN-13: 9780821887509

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This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degeneracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.

Gradient Flows

Gradient Flows PDF

Author: Luigi Ambrosio

Publisher: Springer Science & Business Media

Published: 2008-10-29

Total Pages: 334

ISBN-13: 376438722X

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The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.

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