Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

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Introduction to the Modern Theory of Dynamical Systems

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

Partial Differential Equations Second Edition

We provide a detailed proof of the fact that any open set whose boundary is sufficiently flat in the sense of Reifenberg is also Jones-flat,and hence it admits an extension operator. We discuss various applications of this property, in particular we obtain L 1 estimates for the eigenfunctions of the Laplace operator with Neumann boundary conditions. We also compare different ways of measuring the "distance" between two Reifenberg-flat domains. These results are pivotal to the quantitative stability analysis of the spectrum of the Neumann Laplacian performed in (27).

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On the extension property of reifenberg-flat domains

We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blow-up of the $BV$ norm at the initial time. We apply these results (valid in any space dimension) to the $k\times k$ chromatography system of conservation laws and to the $k\times k$ Keyfitz and Kranzer system, both in one space dimension.

https://www.math.ntnu.no/conservation/2009/015.pdf

Some New Well-Posedness Results for Continuity and Transport Equations, and Applications to the Chromatography System

We give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19–37, 2015), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law. Can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one does not in general have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.

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On the Singular Local Limit for Conservation Laws with Nonlocal Fluxes

We study the limit of the hyperbolic–parabolic approximation 

$$\left\{\begin{array}{l@{\quad}l@{\quad}l} v^{\varepsilon}_t + \tilde{A} \left(v^{\varepsilon}, \, \varepsilon v^{\varepsilon}_x \right) v^{\varepsilon}_x = \varepsilon \tilde{B}(v^{\varepsilon} ) v^{\varepsilon}_{xx} \quad v^{\varepsilon} \in \mathbb{R}^N \cr \tilde{\rm \ss}(v^{\varepsilon} (t, \, 0)) \equiv \bar g \cr v^{\varepsilon} (0, \, x) \equiv \bar{v}_0. \end{array} \right.$$

The function ${\tilde {\ss}}$ is defined in such a way as to guarantee that the initial boundary value problem is well posed even if ${\tilde {B}}$ is not invertible. The data ${\bar {g}}$ and ${\bar {v}_{0}}$ are constant. When ${\tilde {B}}$ is invertible, the previous problem takes the simpler form

$$\left\{\begin{array}{l@{\quad}l@{\quad}l} v^{\varepsilon}_t + \tilde{A}\left(v^{\varepsilon}, \, \varepsilon v^{\varepsilon}_x \right) v^{\varepsilon}_x = \varepsilon \tilde{B}(v^{\varepsilon} ) v^{\varepsilon}_{xx}\quad v^{\varepsilon} \in \mathbb{R}^N \cr v^{\varepsilon} (t, \, 0) \equiv \bar v_b \cr v^{\varepsilon} (0, \, x) \equiv \bar{v}_0. \end{array} \right.$$

Again, the data ${\bar {v}_b}$ and ${\bar {v}_0}$ are constant. The conservative case is included in the previous formulations. Convergence of the ${v^{\varepsilon}}$ , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is, one eigenvalue of ${\tilde {A}}$ can be 0. Second, as pointed out before, we take into account the possibility that ${\tilde {B}}$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if this condition is not satisfied, then pathological behaviors may occur.

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The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation

Consider a nonlocal conservation where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from $0$. We also exhibit a counter-example showing that, if the initial datum attains the value $0$, then there are severe obstructions to a convergence proof.

Laura V. Spinolo

Papers

On the extension property of reifenberg-flat domains

Some New Well-Posedness Results for Continuity and Transport Equations, and Applications to the Chromatography System

On the Singular Local Limit for Conservation Laws with Nonlocal Fluxes

The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation

Local limit of nonlocal traffic models: convergence results and total variation blow-up