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2017 | Book

Integral Inequalities Read chapter Read first chapter

Analytic Inequalities and Their Applications in PDEs

Author: Yuming Qin

Publisher: Springer International Publishing

Book Series : Operator Theory: Advances and Applications

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

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About this book

This book presents a number of analytic inequalities and their applications in partial differential equations. These include integral inequalities, differential inequalities and difference inequalities, which play a crucial role in establishing (uniform) bounds, global existence, large-time behavior, decay rates and blow-up of solutions to various classes of evolutionary differential equations. Summarizing results from a vast number of literature sources such as published papers, preprints and books, it categorizes inequalities in terms of their different properties.

Table of Contents

Frontmatter
Chapter 1. Integral Inequalities
Abstract
It is well known that classical integral inequalities furnishing explicit bounds for an unknown function play a fundamental role in differential and integral equations. In this chapter, we shall first collect some basic integral inequalities.
Yuming Qin
Chapter 2. Differential and Difference Inequalities
Abstract
In this chapter, we establish differential and difference inequalities in analysis that play a role in applications in the subsequent chapters.
Yuming Qin
Chapter 3. Attractors for Evolutionary Differential Equations
Abstract
In this chapter, we prove the existence of global (uniform) attractors for some evolutionary differential equations. The chapter includes four sections. In Section 3.1, we shall use Theorems 1.1.2 and 2.1.3 to establish the existence of global attractors for a nonlinear reaction-diffusion equation.
Yuming Qin
Chapter 4. Global Existence and Uniqueness for Evolutionary PDEs
Abstract
In this chapter, we present some results on global existence and uniqueness of solutions to evolutionary PEDs obtained by application of analytic inequalities in Chapters 1 and 2. This chapter consists of four sections. In Section 4.1, we use the simultaneous singular Bellman–Gronwall inequality, i.e., Theorem 1.3.2, to discuss the local existence, regularity, and continuous dependence on initial data of solutions to a weakly coupled parabolic system for non-regular initial data. In Section 4.2, we use Theorem 1.4.9 to study some properties of solutions to the Cauchy problem for multi-dimensional conservation laws with anomalous diffusion. In Section 4.3, we use Theorem 2.1.19 to investigate the blow-up of solutions of semilinear heat equations.
Yuming Qin
Chapter 5. Global Existence and Uniqueness for Abstract Evolutionary Differential Equations
Abstract
In this chapter, we discuss the global existence and uniqueness for some abstract models and ODEs. The chapter consists of three sections. In Section 5.1, we apply Corollary 1.4.4 to prove the global-in-time existence of solutions to an abstract evolutionary equation written below.
Yuming Qin
Chapter 6. Global Existence and Asymptotic Behavior for Equations of Fluid Dynamics
Abstract
In this chapter, we prove the global existence and asymptotic behavior of solutions to fluid models. The chapter includes five sections. In Section 6.1, we exploit Theorems 1.4.11 and 1.5.1 to show the asymptotic behavior of the solutions to the Navier–Stokes equations in 2D exterior domains.
Yuming Qin
Chapter 7. Asymptotic Behavior of Solutions for Parabolic and Elliptic Equations
Abstract
In this chapter, we shall study the asymptotic behavior for parabolic and elliptic equations. This chapter embraces three sections. In Section 7.1, we shall use Theorems 2.1.14 and 2.3.7 to establish the uniform and decay estimates for flows in a semi-infinite straight channel. In Section 7.2, we shall exploit Theorems 2.3.17–2.3.21 to establish exact rates of convergence for nonlinear PDEs.
Yuming Qin
Chapter 8. Asymptotic Behavior of Solutions to Hyperbolic Equations
Abstract
This chapter mainly studies the asymptotic behavior of solutions to some hyperbolic equations. This chapter includes seven sections. In Section 8.1, we shall use Theorem 2.3.11 to study the decay of solutions to 1D nonlinear wave equations. In Section 8.2, we shall exploit Theorem 2.3.14 to investigate the decay property of the solutions to the initial boundary value problem for a wave equation with a dissipative term. In Section 8.3, we shall apply Theorem 2.3.6 to establish the polynomial decay rate for nonlinear wave equations.
Yuming Qin
Chapter 9. Asymptotic Behavior of Solutions to Thermoviscoelastic, Thermoviscoelastoplastic and Thermomagnetoelastic Equations
Abstract
In this chapter, we shall establish the asymptotic behavior for thermoviscoelastic, thermoviscoelastoplastic and thermomagnetoelastic equations. This chapter consists of three sections. In Section 9.1, we shall first employ Lemma 1.5.4 to extend the decay results in [620] for a viscoelastic system to those for the thermoviscoelastic system (9.1.1) and then establish the existence of the global attractor for the homogeneous thermoviscoelastic system (9.1.54).
Yuming Qin
Chapter 10. Blow-up of Solutions to Nonlinear Hyperbolic Equations and Hyperbolic-Elliptic Inequalities
Abstract
In this chapter, we shall consider the blow-up of solutions to some nonlinear hyperbolic equations and hyperbolic-elliptic inequalities. This chapter consists of seven sections. In Section 10.1, we apply Theorem 2.4.6 to investigate the blow-up of solutions to semilinear wave equations. In Section 10.2, we shall employ Theorem 2.4.22 to study the blow-up of solutions to semilinear wave equations.
Yuming Qin
Chapter 11. Blow-up of Solutions to Abstract Equations and Thermoelastic Equations
Abstract
In this chapter, we shall study the blow-up of solutions to abstract equations and thermoelastic equations. This chapter consists of six sections. In Section 11.1, we shall employ Theorem 2.4.19 to prove the blow-up results of solutions to a class of abstract initial and initial boundary value problems. In Section 11.2, we shall employ Theorem 2.4.20 to study the blow-up of solutions to a class abstract nonlinear equations.
Yuming Qin
Chapter 12. Appendix: Basic Inequalities
Abstract
In this chapter, we shall collect some basic inequalities which play a very crucial role in classical calculus. These inequalities include the Young inequality, the Hölder inequality, the Minkowski inequality, the Jensen inequality, and the Hausdorff–Young inequality, etc.
Yuming Qin
Backmatter
Metadata
Title
Analytic Inequalities and Their Applications in PDEs
Author
Yuming Qin
Copyright Year
2017
Electronic ISBN
978-3-319-00831-8
Print ISBN
978-3-319-00830-1
DOI
https://doi.org/10.1007/978-3-319-00831-8

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