Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors

This book is designed to present some recent results on some nonlinear parabolic-hyp- bolic coupled systems arising from physics, mechanics and material science such as the compressible Navier-Stokes equations, thermo(visco)elastic systems and elastic systems. Some of the content of this book is based on research carried out by the author and his collaborators in recent years. Most of it has been previously published only in original papers,andsomeofthematerialhasneverbeenpublisheduntilnow.Therefore,theauthor hopes that the book will bene?t both the interested beginner in the ?eld and the expert. AllthemodelsunderconsiderationinChapters2-10arebuiltonnonlinearevolution equations that are parabolic-hyperbolic coupled systems of partial differential equations with time t as one of the independentvariables. This type of partial differential equations arises not only in many ?elds of mathematics, but also in other branches of science such as physics, mechanics and materials science, etc. For example, some models studied in this book, such as the compressible Navier-Stokes equations (a 1D heat conductive v- cous real gas and a polytropic ideal gas) from ?uid mechanics, and thermo(visco)elastic systemsfrommaterialsscience, are typicalexamplesof nonlinearevolutionaryequations. It is well known that the properties of solutions to nonlinear parabolic-hyperbolic coupledsystems are very different from those of parabolicor hyperbolicequations. Since the 1970s,more andmore mathematicianshave begunto focustheir interests onthe study of local well-posedness, global well-posedness and blow-up of solutions in a ?nite time.

Résumé

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.

Contenu
Preliminary.- A One-dimensional Nonlinear Viscous and Heat-conductive Real Gas.- A One-dimensional Polytropic Viscous and Heat-conductive Gas.- A Polytropic Ideal Gas in Bounded Annular Domains in .- A Polytropic Viscous Gas with Cylinder Symmetry in .- One-dimensional Nonlinear Thermoviscoelasticity.- A Nonlinear One-dimensional Thermoelastic System with a Thermal Memory.- One-dimensional Thermoelastic Equations of Hyperbolic Type.- Blow-up for the Cauchy Problem in Nonlinear One-dimensional Thermoelasticity.- Large-Time Behavior of Energy in Multi-Dimensional Elasticity.