Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2018 | Book

Qualitative Methods for Classes of Nonlinear Systems: Constructive Existence Results Read chapter Read first chapter

Qualitative and Quantitative Analysis of Nonlinear Systems

Theory and Applications

Authors: Michael Z. Zgurovsky, Pavlo O. Kasyanov

Publisher: Springer International Publishing

Book Series : Studies in Systems, Decision and Control

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

Here, the authors present modern methods of analysis for nonlinear systems which may occur in fields such as physics, chemistry, biology, or economics. They concentrate on the following topics, specific for such systems:

(a) constructive existence results and regularity theorems for all weak solutions;

(b) convergence results for solutions and their approximations;

(c) uniform global behavior of solutions in time; and

(d) pointwise behavior of solutions for autonomous problems with possible gaps by the phase variables. The general methodology for the investigation of dissipative dynamical systems with several applications including nonlinear parabolic equations of divergent form, nonlinear stochastic equations of parabolic type, unilateral problems, nonlinear PDEs on Riemannian manifolds with or without boundary, contact problems as well as particular examples is established. As such, the book is addressed to a wide circle of mathematical, mechanical and engineering readers.

Table of Contents

Frontmatter

Existence and Regularity Results, Quantitative Methods and Their Convergence

Frontmatter
Chapter 1. Qualitative Methods for Classes of Nonlinear Systems: Constructive Existence Results
Abstract
In this chapter we establish the existence results for classes of nonlinear systems. Section 2.​1 devoted to the first order differential-operator equations and inclusions. In Sect. 2.​2 we consider the second order operator differential equations and inclusions in special classes of infinite-dimensional spaces of distributions. Section 2.​3 devoted to the existence of strong solutions for evolutional variational inequalities with nonmonotone potential. The penalty method for strong solutions is justified. A nonlinear parabolic equations of divergent form are considered as examples of applications in Sect. 2.​4.
Michael Z. Zgurovsky, Pavlo O. Kasyanov
Chapter 2. Regularity of Solutions for Nonlinear Systems
Abstract
In this chapter we establish sufficient conditions for regularity of all weak solutions for nonlinear systems. We note that the respective Cauchy problems may have nonunique weak solution. In Sect. 2.1 we establish regularity of all weak solutions for parabolic feedback control problems. Section 2.2 devoted to artificial control method for nonlinear partial differential equations and inclusions. The regularity of all weak solutions is obtained. In Sect. 2.3 we consider regularity results of all weak solutions for nonlinear reaction-diffusion systems with nonlinear growth. In Sect. 2.4 we consider the following examples of applications: a parabolic feedback control problem; a model of conduction of electrical impulses in nerve axons; a climate energy balance model; FitzHugh–Nagumo System; a model of combustion in porous media.
Michael Z. Zgurovsky, Pavlo O. Kasyanov
Chapter 3. Advances in the 3D Navier-Stokes Equations
Abstract
In this chapter we provide a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data. Moreover, we establish sufficient conditions for Leray-Hopf property of a weak solution for the 3D Navier-Stokes system. Under such conditions this weak solution is rightly continuous in the standard phase space H endowed with the strong convergence topology.
Michael Z. Zgurovsky, Pavlo O. Kasyanov

Convergence Results in Strongest Topologies

Frontmatter
Chapter 4. Strongest Convergence Results for Weak Solutions of Non-autonomous Reaction-Diffusion Equations with Carathéodory’s Nonlinearity
Abstract
In this chapter we consider the problem of uniform convergence results for all globally defined weak solutions of non-autonomous reaction-diffusion system with Carathéodory’s nonlinearity satisfying standard sign and polynomial growth assumptions. The main contributions of this chapter are: the uniform convergence results for all globally defined weak solutions of non-autonomous reaction-diffusion equations with Carathéodory’s nonlinearity and sufficient conditions for the convergence of weak solutions in strongest topologies.
Michael Z. Zgurovsky, Pavlo O. Kasyanov
Chapter 5. Strongest Convergence Results for Weak Solutions of Feedback Control Problems
Abstract
In this chapter we establish strongest convergence results for weak solutions of feedback control problems. In Sect. 5.1 we set the problem. Section 5.2 devoted to the regularity of all weak solutions and their additional properties. In Sect. 5.3 we consider convergence of weak solutions results in the strongest topologies. As examples of applications we consider a model of combustion in porous media; a model of conduction of electrical impulses in nerve axons; and a climate energy balance model.
Michael Z. Zgurovsky, Pavlo O. Kasyanov
Chapter 6. Strongest Convergence Results for Weak Solutions of Differential-Operator Equations and Inclusions
Abstract
In this chapter we establish strongest convergence results for weak solutions of differential-operator equations and inclusions. In Sect. 6.1 we consider first order differential-operator equations and inclusions. Section 6.2 devoted to convergence results for weak solutions of second order operator differential equations and inclusions. In Sect. 6.3 we consider the following examples of applications: nonlinear parabolic equations of divergent form; nonlinear problems on manifolds with and without boundary: a climate energy balance model; a model of conduction of electrical impulses in nerve axons; viscoelastic problems with nonlinear “reaction-displacement” law.
Michael Z. Zgurovsky, Pavlo O. Kasyanov

Uniform Global Behavior of Solutions: Uniform Attractors, Flattening and Entropy

Frontmatter
Chapter 7. Uniform Global Attractors for Non-autonomous Dissipative Dynamical Systems
Abstract
In this chapter we consider sufficient conditions for the existence of uniform compact global attractor for non-autonomous dynamical systems in special classes of infinite-dimensional phase spaces. The obtained generalizations allow us to avoid the restrictive compactness assumptions on the space of shifts of non-autonomous terms in particular evolution problems. The results are applied to several evolution inclusions.
Michael Z. Zgurovsky, Pavlo O. Kasyanov
Chapter 8. Uniform Trajectory Attractors for Non-autonomous Nonlinear Systems
Abstract
In this chapter we study uniform trajectory attractors for non-autonomous nonlinear systems. In Sect. 8.1 we establish the existence of uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathéodory’s nonlinearity. Section 8.2 devoted to structural properties of the uniform global attractor for non-autonomous reaction-diffusion system in which uniqueness of Cauchy problem is not guarantied. In the case of translation compact time-depended coefficients it is established that the uniform global attractor consists of bounded complete trajectories of corresponding multi-valued processes. Under additional sign conditions on non-linear term we also prove (and essentially use previous result) that the uniform global attractor is, in fact, bounded set in \(L^{\infty }(\varOmega )\cap H_0^1(\varOmega )\). Section 8.3 devoted to uniform trajectory attractors for nonautonomous dissipative dynamical systems. As applications we may consider FitzHugh–Nagumo system (signal transmission across axons), complex Ginzburg–Landau equation (theory of superconductivity), Lotka–Volterra system with diffusion (ecology models), Belousov–Zhabotinsky system (chemical dynamics) and many other reaction-diffusion type systems from Sect. 2.​4.
Michael Z. Zgurovsky, Pavlo O. Kasyanov
Chapter 9. Indirect Lyapunov Method for Autonomous Dynamical Systems
Abstract
In this chapter we establish indirect Lyapunov method for autonomous dynamical systems. Section 9.1 devoted to the first order autonomous differential-operator equations and inclusions. In Sect. 9.2 we consider the second order autonomous operator differential equations and inclusions. In Sect. 9.3 we examine examples of applications. In particular, a model of combustion in porous media; a model of conduction of electrical impulses in nerve axons; viscoelastic problems with nonlinear “reaction-displacement” law etc.
Michael Z. Zgurovsky, Pavlo O. Kasyanov
Backmatter
Metadata
Title
Qualitative and Quantitative Analysis of Nonlinear Systems
Authors
Michael Z. Zgurovsky
Pavlo O. Kasyanov
Copyright Year
2018
Electronic ISBN
978-3-319-59840-6
Print ISBN
978-3-319-59839-0
DOI
https://doi.org/10.1007/978-3-319-59840-6

Premium Partner

    Image Credits