This 2006 book details exact solutions to the Navier-Stokes equations for senior undergraduates and graduates or research reference.

This volume collects the articles presented at the Third International Conference on ?The Navier-Stokes Equations: Theory and Numerical Methods?, held in Oberwolfach, Germany. The articles are important contributions to a wide variety of topics in the Navier-Stokes theory: general boundary conditions, flow exterior to an obstacle, conical boundary points, the controllability of solutions, compressible flow, non-Newtonian flow, magneto-hydrodynamics, thermal convection, the interaction of fluids with elastic solids, the regularity of solutions, and Rothe's method of approximation.

These proceedings contain original (refereed) research articles by specialists from many countries, on a wide variety of aspects of Navier-Stokes equations. Additionally, 2 survey articles intended for a general readership are included: one surveys the present state of the subject via open problems, and the other deals with the interplay between theory and numerical analysis.

A clear and focused guide to the Navier-Stokes equations that govern fluid motion, including exercises and fully worked solutions.

The primary objective of this monograph is to develop an elementary and se- containedapproachtothemathematicaltheoryofaviscousincompressible?uid n in a domain ? of the Euclidean spaceR , described by the equations of Navier- Stokes. The book is mainly directed to students familiar with basic functional analytic tools in Hilbert and Banach spaces. However, for readers’ convenience, in the ?rst two chapters we collect, without proof some fundamental properties of Sobolev spaces, distributions, operators, etc. Another important objective is to formulate the theory for a completely general domain ?. In particular, the theory applies to arbitrary unbounded, non-smooth domains. For this reason, in the nonlinear case, we have to restrict ourselves to space dimensions n=2,3 that are also most signi?cant from the physical point of view. For mathematical generality, we will develop the l- earized theory for all n? 2. Although the functional-analytic approach developed here is, in principle, known to specialists, its systematic treatment is not available, and even the diverseaspectsavailablearespreadoutintheliterature.However,theliterature is very wide, and I did not even try to include a full list of related papers, also because this could be confusing for the student. In this regard, I would like to apologize for not quoting all the works that, directly or indirectly, have inspired this monograph.

The book provides a comprehensive, detailed and self-contained treatment of the fundamental mathematical properties of boundary-value problems related to the Navier-Stokes equations. These properties include existence, uniqueness and regularity of solutions in bounded as well as unbounded domains. Whenever the domain is unbounded, the asymptotic behavior of solutions is also investigated. This book is the new edition of the original two volume book, under the same title, published in 1994. In this new edition, the two volumes have merged into one and two more chapters on steady generalized oseen flow in exterior domains and steady Navier–Stokes flow in three-dimensional exterior domains have been added. Most of the proofs given in the previous edition were also updated. An introductory first chapter describes all relevant questions treated in the book and lists and motivates a number of significant and still open questions. It is written in an expository style so as to be accessible also to non-specialists.Each chapter is preceded by a substantial, preliminary discussion of the problems treated, along with their motivation and the strategy used to solve them. Also, each chapter ends with a section dedicated to alternative approaches and procedures, as well as historical notes. The book contains more than 400 stimulating exercises, at different levels of difficulty, that will help the junior researcher and the graduate student to gradually become accustomed with the subject. Finally, the book is endowed with a vast bibliography that includes more than 500 items. Each item brings a reference to the section of the book where it is cited. The book will be useful to researchers and graduate students in mathematics in particular mathematical fluid mechanics and differential equations. Review of First Edition, First Volume: “The emphasis of this book is on an introduction to the mathematical theory of the stationary Navier-Stokes equations. It is written in the style of a textbook and is essentially self-contained. The problems are presented clearly and in an accessible manner. Every chapter begins with a good introductory discussion of the problems considered, and ends with interesting notes on different approaches developed in the literature. Further, stimulating exercises are proposed. (Mathematical Reviews, 1995)

Contents: A New Approach to the Helmholtz Decomposition and the Neumann Problem in Lq-Spaces for Bounded and Exterior Domains (C G Simader & H Sohr)On the Energy Equation and on the Uniqueness for D-Solutions to Steady Navier-Stokes Equations in Exterior Domains (G P Galdi)On the Asymptotic Structure of D-Solutions to Steady Navier-Stokes Equations in Exterior Domains (G P Galdi)On the Solvability of an Evolution Free Boundary Problem for the Navier-Stokes Equation in Hölder Spaces of Functions (I S Mogilevskii & V A Solonnikov) Readership: Applied mathematicians.

This book collects together a unique set of articles dedicated to several fundamental aspects of the Navier–Stokes equations. As is well known, understanding the mathematical properties of these equations, along with their physical interpretation, constitutes one of the most challenging questions of applied mathematics. Indeed, the Navier-Stokes equations feature among the Clay Mathematics Institute's seven Millennium Prize Problems (existence of global in time, regular solutions corresponding to initial data of unrestricted magnitude). The text comprises three extensive contributions covering the following topics: (1) Operator-Valued H∞-calculus, R-boundedness, Fourier multipliers and maximal Lp-regularity theory for a large, abstract class of quasi-linear evolution problems with applications to Navier–Stokes equations and other fluid model equations; (2) Classical existence, uniqueness and regularity theorems of solutions to the Navier–Stokes initial-value problem, along with space-time partial regularity and investigation of the smoothness of the Lagrangean flow map; and (3) A complete mathematical theory of R-boundedness and maximal regularity with applications to free boundary problems for the Navier–Stokes equations with and without surface tension. Offering a general mathematical framework that could be used to study fluid problems and, more generally, a wide class of abstract evolution equations, this volume is aimed at graduate students and researchers who want to become acquainted with fundamental problems related to the Navier–Stokes equations.

This monograph considers the motion of incompressible fluids described by the Navier-Stokes equations with large inflow and outflow, and proves the existence of global regular solutions without any restrictions on the magnitude of the initial velocity, the external force, or the flux. To accomplish this, some assumptions are necessary: The flux is close to homogeneous, and the initial velocity and the external force do not change too much along the axis of the cylinder. This is achieved by utilizing a sophisticated method of deriving energy type estimates for weak solutions and global estimates for regular solutions—an approach that is wholly unique within the existing literature on the Navier-Stokes equations. To demonstrate these results, three main steps are followed: first, the existence of weak solutions is shown; next, the conditions guaranteeing the regularity of weak solutions are presented; and, lastly, global regular solutions are proven. This volume is ideal for mathematicians whose work involves the Navier-Stokes equations, and, more broadly, researchers studying fluid mechanics.

V.A. Solonnikov, A. Tani: Evolution free boundary problem for equations of motion of viscous compressible barotropic liquid.- W. Borchers, T. Miyakawa:On some coercive estimates for the Stokes problem in unbounded domains.- R. Farwig, H. Sohr: An approach to resolvent estimates for the Stokes equations in L(q)-spaces.- R. Rannacher: On Chorin's projection method for the incompressible Navier-Stokes equations.- E. S}li, A. Ware: Analysis of the spectral Lagrange-Galerkin method for the Navier-Stokes equations.- G. Grubb: Initial value problems for the Navier-Stokes equations with Neumann conditions.- B.J. Schmitt, W. v.Wahl: Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the Boussinesq-equations.- O. Walsh: Eddy solutions of the Navier-Stokesequations.- W. Xie: On a three-norm inequality for the Stokes operator in nonsmooth domains.