Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations
Title | Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations PDF eBook |
Author | Maria Colombo |
Publisher | Springer |
Pages | 285 |
Release | 2017-06-07 |
Genre | Mathematics |
ISBN | 8876426078 |
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
Title | Weighted Sobolev Spaces and Degenerate Elliptic Equations PDF eBook |
Author | Albo Carlos Cavalheiro |
Publisher | Cambridge Scholars Publishing |
Pages | 333 |
Release | 2023-09-29 |
Genre | Mathematics |
ISBN | 1527551679 |
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
Title | Spaces of Measures and their Applications to Structured Population Models PDF eBook |
Author | Christian Düll |
Publisher | Cambridge University Press |
Pages | 322 |
Release | 2021-10-07 |
Genre | Mathematics |
ISBN | 1009020471 |
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.
Existence and Uniqueness of Maximal Regular Flows for Non-smooth Vector Fields
Title | Existence and Uniqueness of Maximal Regular Flows for Non-smooth Vector Fields PDF eBook |
Author | Dario Koch |
Publisher | |
Pages | |
Release | 2016 |
Genre | |
ISBN |
The Flow Associated to Weakly Differentiable Vector Fields
Title | The Flow Associated to Weakly Differentiable Vector Fields PDF eBook |
Author | Gianluca Crippa |
Publisher | Edizioni della Normale |
Pages | 0 |
Release | 2009-03-27 |
Genre | Mathematics |
ISBN | 9788876423406 |
The aim of this book is to provide a self-contained introduction and an up-to-date survey on many aspects of the theory of transport equations and ordinary differential equations with non-smooth velocity fields. The interest in this topic is motivated by important issues in nonlinear PDEs, in particular conservation laws and fluid mechanics. A fascinating feature of this research area, which is currently of concern in mathematics, is the interplay between PDE techniques and geometric measure theory techniques. Several masterpieces appear in the related literature, balancing completely rigorous proofs with more heuristic arguments. A consistent part of the book is based on results obtained by the author in collaboration with other mathematicians. After a short introduction to the classical smooth theory, the book is divided into two parts. The first part focuses on the PDE aspect of the problem, presenting some general tools of this analysis, many well-posedness results, an abstract characterization of the well-posedness, and some examples showing the sharpness of the assumptions made. The second part, instead, deals with the ODE aspect of the problem, respectively by an abstract connection with the PDE, and by some direct and simple (but powerful) a priori estimates.
Degenerate Elliptic Equations
Title | Degenerate Elliptic Equations PDF eBook |
Author | Serge Levendorskii |
Publisher | Springer Science & Business Media |
Pages | 442 |
Release | 2013-11-11 |
Genre | Mathematics |
ISBN | 9401712158 |
This volume is the first to be devoted to the study of various properties of wide classes of degenerate elliptic operators of arbitrary order and pseudo-differential operators with multiple characteristics. Conditions for operators to be Fredholm in appropriate weighted Sobolev spaces are given, a priori estimates of solutions are derived, inequalities of the Grding type are proved, and the principal term of the spectral asymptotics for self-adjoint operators is computed. A generalization of the classical Weyl formula is proposed. Some results are new, even for operators of the second order. In addition, an analogue of the Boutet de Monvel calculus is developed and the index is computed. For postgraduate and research mathematicians, physicists and engineers whose work involves the solution of partial differential equations.
Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order
Title | Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order PDF eBook |
Author | A. V. Ivanov |
Publisher | American Mathematical Soc. |
Pages | 306 |
Release | 1984 |
Genre | Mathematics |
ISBN | 9780821830802 |