Hybrid Function Spaces, Heat and Navier-Stokes Equations

This book is the continuation of Local Function Spaces, Heat and Navier-Stokes Equations (EMS Tracts in Mathematics, volume 20, 2013) by the author. A new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and Holder-Zygmund spaces on the one hand and Morrey-Campanato spaces on the other. Morrey-Campanato spaces extend the notion of functions of bounded mean oscillation. These spaces play a crucial role in the theory of linear and nonlinear PDEs. Chapter 1...
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This book is the continuation of Local Function Spaces, Heat and Navier-Stokes Equations (EMS Tracts in Mathematics, volume 20, 2013) by the author. A new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and Holder-Zygmund spaces on the one hand and Morrey-Campanato spaces on the other. Morrey-Campanato spaces extend the notion of functions of bounded mean oscillation. These spaces play a crucial role in the theory of linear and nonlinear PDEs. Chapter 1 (Introduction) describes the main motivations and intentions of this book. Chapter 2 is a self-contained introduction to Morrey spaces. Chapter 3 deals with hybrid smoothness spaces (which are between local and global spaces) in Euclidean $n$-space based on the Morrey-Campanato refinement of the Lebesgue spaces. The presented approach, which relies on wavelet decompositions, is applied in Chapter 4 to linear and nonlinear heat equations in global and hybrid spaces. The obtained assertions about function spaces and nonlinear heat equations are used in Chapters 5 and 6 to study Navier-Stokes equations in hybrid and global spaces. This book is addressed to graduate students and mathematicians who have a working knowledge of basic elements of (global) function spaces and who are interested in applications to nonlinear PDEs with heat and Navier-Stokes equations as prototypes.