Gregory Seregin

This book is based on the lecture notes for the TCC (Taught Course Centre for graduates) course given by the author in Trinity Terms of 2009-2011 at the Mathematical Institute of Oxford University. It provides an accessible yet rigorous introduction to the mathematical theory of the Navier-Stokes equations, including both classical PDE theory and modern regularity results developed in the style of the St Petersburg school. The book covers fundamental concepts from basic theory to state-of-the-art results, with a focus on the interplay between regularity and well-posedness — a central theme in the study of the Navier-Stokes equations and one of the Millennium Prize Problems. The second edition introduces major new material that extends the scope of the original text. Chapter 8 explores the regularity of axially symmetric solutions and examines Type I and Type II blowup in suitable weak solutions, offering insights into possible singularity formation and the broader global regularity problem. In addition, Appendix C provides detailed proofs of key results, enhancing the mathematical rigor and connecting the material to ongoing research and open problems in fluid dynamics. Together, the comprehensive coverage of classical and modern theory, enriched with these new contributions, makes this edition a valuable resource for graduate students, researchers, and anyone interested in the analytical foundations of fluid dynamics.

The lecture notes in this book are based on the TCC (Taught Course Centre for graduates) course given by the author in Trinity Terms of 2009-2011 at the Mathematical Institute of Oxford University. It contains more or less an elementary introduction to the mathematical theory of the Navier-Stokes equations as well as the modern regularity theory for them. The latter is developed by means of the classical PDE's theory in the style that is quite typical for St Petersburg's mathematical school of the Navier-Stokes equations. The global unique solvability (well-posedness) of initial boundary value problems for the Navier-Stokes equations is in fact one of the seven Millennium problems stated by the Clay Mathematical Institute in 2000. It has not been solved yet. However, a deep connection between regularity and well-posedness is known and can be used to attack the above challenging problem. This type of approach is not very well presented in the modern books on the mathematical theory of the Navier-Stokes equations. Together with introduction chapters, the lecture notes will be a self-contained account on the topic from the very basic stuff to the state-of-art in the field.

This volume brings together five contributions to mathematical fluid mechanics, a classical but still very active research field which overlaps with physics and engineering. The contributions cover not only the classical Navier-Stokes equations for an incompressible Newtonian fluid, but also generalized Newtonian fluids, fluids interacting with particles and with solids, and stochastic models. The questions addressed in the lectures range from the basic problems of existence of weak and more regular solutions, the local regularity theory and analysis of potential singularities, qualitative and quantitative results about the behavior in special cases, asymptotic behavior, statistical properties and ergodicity.

Variational methods are applied to prove the existence of weak solutions for boundary value problems from the deformation theory of plasticity as well as for the slow, steady state flow of generalized Newtonian fluids including the Bingham and Prandtl-Eyring model. For perfect plasticity the role of the stress tensor is emphasized by studying the dual variational problem in appropriate function spaces. The main results describe the analytic properties of weak solutions, e.g. differentiability of velocity fields and continuity of stresses. The monograph addresses researchers and graduate students interested in applications of variational and PDE methods in the mechanics of solids and fluids.