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Analytic Inequalities and Their Applications in PDEs
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.
1D Radiative Fluid and Liquid Crystal Equations
This book presents recent results on nonlinear evolutionary fluid equations, in particular the global well-posedness and asymptotic behavior of solutions to 1D radiative fluid equations, as well as liquid crystal equations. Most of the material in this book was prepared by the author over the past few years. This book has two main features. Firstly, there are more known results on higher dimensional radiative fluid systems but only on the local existence and explosion of solutions; while the existing findings on the one-dimensional case present some shortcomings, this book introduces corrections and improvements of these shortcomings. Secondly, the current findings on the high-dimensional compressible liquid crystal fluid equations are few and include only globally existing solutions but not the asymptotic behavior of the solutions; the author developed not only the global existence and regularity of the solutions, but also the asymptotic behavior of the solutions for the one-dimensional case in the chapter 3 of this book. Therefore, this work provides the reader with complete elements related to the one-dimensional compressible liquid crystal fluid system.
Attractors for Nonlinear Autonomous Dynamical Systems
This book introduces complete and systematic theories of infinite-dimensional dynamical systems and their applications in partial differential equations, especially in the models of fluid mechanics. It is based on the first author’s lecture “Infinite dimensional dynamical systems on nonlinear autonomous systems” given to graduate students in Donghua University since 2004. This book presents recent results that have been carried out by the authors on autonomous nonlinear evolutionary equations arising from physics, fluid mechanics and material science such as the Navier–Stokes equations, Navier–Stokes–Voight systems, the nonlinear thermoviscoelastic system, etc.
Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems
This book presents recent results on nonlinear parabolic-hyperbolic coupled systems such as the compressible Navier-Stokes equations, and liquid crystal system. It summarizes recently published research by the authors and their collaborators, but also includes new and unpublished material. All models under consideration are built on compressible equations and liquid crystal systems. This type of partial differential equations arises not only in many fields of mathematics, but also in other branches of science such as physics, fluid dynamics and material science.
Global Well-Posedness for Some Fluid Models
This book presents recent results on global well-posedness including asymptotic behavior of global solutions to some fluid models, such as combustion model of radiative gas, radiation hydrodynamics model, Navier-Stokes equations with capillary and p-th power Newtonian fluid model. These models have the similar structures, which consist of Navier-Stokes equations coupled with other equation or with other effects. Results collected in this book are established by the authors and their collaborators in recent years.
Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors
This book presents recent results concerning the global existence in time, the large-time behavior, decays of solutions and the existence of global attractors for nonlinear parabolic-hyperbolic coupled systems of evolutionary partial differential equations.
Global Existence and Uniqueness of Nonlinear Evolutionary Fluid Equations
This book presents recent results on nonlinear evolutionary fluid equations such as the compressible (radiative) magnetohydrodynamics (MHD) equations, compressible viscous micropolar fluid equations, the full non-Newtonian fluid equations and non-autonomous compressible Navier-Stokes equations. These types of partial differential equations arise in many fields of mathematics, but also in other branches of science such as physics and fluid dynamics. This book will be a valuable resource for graduate students and researchers interested in partial differential equations, and will also benefit practitioners in physics and engineering.
Dizang Qu
Yongle Town is located at the foot of Yunwu Mountain. There are about 600 families in the town, which is the largest town within 800 miles of Yunwu Mountain.
Integral and Discrete Inequalities and Their Applications
This book concentrates on one- and multi-dimensional nonlinear integral and discrete Gronwall-Bellman type inequalities. It complements the author’s book on linear inequalities and serves as an essential tool for researchers interested in differential (ODE and PDE), difference, and integral equations. The present volume is part 2 of the author’s two-volume work on inequalities. Integral and discrete inequalities are a very important tool in classical analysis and play a crucial role in establishing the well-posedness of the related equations, i.e., differential, difference and integral equations.
Global Well-posedness and Asymptotic Behavior of the Solutions to Non-classical Thermo(visco)elastic Models
This book presents recent findings on the global existence, the uniqueness and the large-time behavior of global solutions of thermo(vis)coelastic systems and related models arising in physics, mechanics and materials science such as thermoviscoelastic systems, thermoelastic systems of types II and III, as well as Timoshenko-type systems with past history. Part of the book is based on the research conducted by the authors and their collaborators in recent years. The book will benefit interested beginners in the field and experts alike.
