This second edition, like the first, attempts to arrive as simply as possible at some central problems in the Navier–Stokes equations in the following areas: existence, uniqueness, and regularity of solutions in space dimensions two and three; large time behavior of solutions and attractors; and numerical analysis of the Navier–Stokes equations. Since publication of the first edition of these lectures in 1983, there has been extensive research in the area of inertial manifolds for Navier–Stokes equations. These developments are addressed in a new section devoted entirely to inertial manifolds.Inertial manifolds were first introduced under this name in 1985 and, since then, have been systematically studied for partial differential equations of the Navier–Stokes type. Inertial manifolds are a global version of central manifolds. When they exist they encompass the complete dynamics of a system, reducing the dynamics of an infinite system to that of a smooth, finite-dimensional one called the inertial system. Although the theory of inertial manifolds for Navier–Stokes equations is not complete at this time, there is already a very interesting and significant set of results which deserves to be known, in the hope that it will stimulate further research in this area. These results are reported in this edition.Part I presents the Navier–Stokes equations of viscous incompressible fluids and the main boundary-value problems usually associated with these equations.
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The case of the flow in a bounded domain with periodic or zero boundary conditions is studied and the functional setting of the equation as well as various results on existence, uniqueness, and regularity of time-dependent solutions are given. Part II studies the behavior of solutions of the Navier–Stokes equation when t approaches infinity and attempts to explain turbulence. Part III treats questions related to numerical approximation. In the Appendix, which is new to the second edition, concepts of inertial manifolds are described, definitions and some typical results are recalled, and the existence of inertial systems for two-dimensional Navier–Stokes equations is shown. Since publication of the first edition of this book in 1983, a very active area in the theory of Navier–Stokes equations has been the study of these equations as a dynamical system in relation to the dynamical system approach to turbulence.
A large number of results have been derived concerning the long-time behavior of the solutions, the attractors for the Navier–Stokes equations and their approximation, the problem of the existence of exact inertial manifolds and approximate inertial manifolds, and new numerical algorithms stemming from dynamical systems theory, such as the nonlinear Galerkin method. Numerical simulations of turbulence and other numerical methods based on different approaches have also been studied intensively during this decade.Most of the results presented in the first edition of this book are still relevant; they are not altered here. Recent results on the numerical approximation of the Navier–Stokes equation or the study of the dynamical system that they generate are addressed thoroughly in more specialized publications.In addition to some minor alterations, the second edition of Navier–Stokes Equations and Nonlinear Functional Analysis has been updated by the addition of a new appendix devoted to inertial manifolds for Navier–Stokes equations. In keeping with the spirit of these notes, which was to arrive as rapidly and as simply as possible at some central problems in the Navier–Stokes equations, we choose to add this section addressing one of the topics of extensive research in recent years.Although some related concepts and results had existed earlier, inertial manifolds were first introduced under this name in 1985 and systematically studied for partial differential equations of the Navier–Stokes type since that date. At this time the theory of inertial manifolds for Navier–Stokes equations is not complete, but there is already available a set of results which deserves to be known, in the hope that this will stimulate further research in this area.Inertial manifolds are a global version of central manifolds. When they exist they encompass the complete dynamics of a system, reducing the dynamics of an infinite system to that of a smooth, finite-dimensional one called the inertial system.
In the Appendix we describe the concepts and recall the definitions and some typical results; we show the existence of inertial manifolds for the Navier–Stokes equations with an enhanced viscosity.