Qin Y., Liu X., Wang T. Global Existence and Uniqueness of Nonlinear Evolutionary Fluid Equations

Springer, 2015. — 217 p.
This book is aimed at presenting some recent results on global wellposedness of nonlinear evolutionary fluid equations. Most of the material of this book is based on the research carried out by the authors and their collaborators in recent years. Some of the material has been previously published only in original papers, while some of it has never been published until now.
There are 8 chapters in this book. Chapter 1 concerns the global existence and asymptotic behavior of solutions to a 1D magnetohydrodynamics (MHD) fluid system. Chapter 2 concerns the global existence and exponential stability of solutions to a 1D compressible and radiative MHD flow model. In Chapter 3 we study the global existence and exponential stability of solutions to a 1D thermally radiative MHD with self-gravitation. Chapter 4 investigates the global existence of solutions for a 1D self-gravitating viscous radiative and reactive gas model. Chapter 5 deals with the global existence and exponential stability of solutions to a compressible viscous micropolar fluid model. Chapter 6 will deal with the global existence and exponential stability of solutions to a compressible viscous micropolar fluid model. In Chapter 7 we establish the global existence and exponential stability of solutions to a full non-Newtonian fluid model (p 2), which is very different from those Newtonian fluid models (p = 2) discussed in Chapters 3–5 in Qin and Huang [102]. Moreover, Chapter 5 in Qin and Huang [102] mainly deals with a model of Newtonian radiative fluids where the radiative effect is accounted for with different constitutive relations from those in Chapter 7 in this book, where only the non-Newtonian fluid without radiative effect is considered. To deal with such a non-Newtonian model, we need to design more delicate and more complicated estimates than those for the Newtonian model in Chapters 3–5 in Qin and Huang [102]. Chapter 8 is a continuation of Chapter 1 in Qin and Huang [102], in which the global existence of solutions in Hi (i = 1, 2, 4) has been obtained. In this chapter, we further establish the exponential stability of spherically symmetric solutions for nonlinear non-autonomous compressible Navier-Stokes equations based on the uniform estimates derived in Chapter 1 of Qin and Huang [102].